JEE Main & Advanced Mathematics Indefinite Integrals Question Bank

done Integration by Substitution Total Questions - 148

• question_answer1)
  \[\int_{{}}^{{}}{\frac{dx}{1+{{e}^{x}}}=}\] [MP PET 1991; Roorkee 1977]
  A)
             \[\log (1+{{e}^{x}})\] done clear
  B)
             \[-\log (1+{{e}^{-x}})\] done clear
  C)
             \[-\log (1-{{e}^{-x}})\] done clear
  D)
             \[\log ({{e}^{-x}}+{{e}^{-2x}})\] done clear
  View Solution play_arrow

• question_answer2)
  \[\int_{{}}^{{}}{\frac{dx}{{{e}^{x}}+{{e}^{-x}}}=}\] [Bihar CEE 1976; MNR 1974]
  A)
             \[{{\tan }^{-1}}({{e}^{-x}})\] done clear
  B)
             \[{{\tan }^{-1}}({{e}^{x}})\] done clear
  C)
             \[\log ({{e}^{x}}-{{e}^{-x}})\] done clear
  D)
             \[\log ({{e}^{x}}+{{e}^{-x}})\] done clear
  View Solution play_arrow

• question_answer3)
  \[\int_{{}}^{{}}{\frac{{{e}^{\sqrt{x}}}\cos {{e}^{\sqrt{x}}}}{\sqrt{x}}dx}=\]
  A)
             \[2\sin {{e}^{\sqrt{x}}}\] done clear
  B)
             \[\sin {{e}^{\sqrt{x}}}\] done clear
  C)
             \[2\cos {{e}^{\sqrt{x}}}\] done clear
  D)
             \[-2\sin {{e}^{\sqrt{x}}}\] done clear
  View Solution play_arrow

• question_answer4)
  \[\int_{{}}^{{}}{\frac{dx}{x+x\log x}=}\]    [MP PET 1993; Roorkee 1977]
  A)
             \[\log (1+\log x)\] done clear
  B)
             \[\log \log (1+\log x)\] done clear
  C)
             \[\log x+\log (\log x)\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer5)
  To find the value of \[\int_{{}}^{{}}{\frac{1+\log x}{x}\text{ }}dx\], the proper substitution is           [MP PET 1988]
  A)
             \[\log x=t\] done clear
  B)
             \[1+\log x=t\] done clear
  C)
             \[\frac{1}{x}=t\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer6)
  \[\int_{{}}^{{}}{\frac{\sec x\ dx}{\sqrt{\cos 2x}}}=\]
  A)
             \[{{\sin }^{-1}}(\tan x)\] done clear
  B)
             \[\tan x\] done clear
  C)
             \[{{\cos }^{-1}}(\tan x)\] done clear
  D)
             \[\frac{\sin x}{\sqrt{\cos x}}\] done clear
  View Solution play_arrow

• question_answer7)
  To find the value of \[\int_{{}}^{{}}{\frac{dx}{x\sqrt{2ax-{{x}^{2}}}}}\], the suitable substitution is
  A)
             \[x=a\cos t\] done clear
  B)
             \[x=2a\cos t\] done clear
  C)
             \[x=2at\] done clear
  D)
             \[x=2a{{\sin }^{2}}t\] done clear
  View Solution play_arrow

• question_answer8)
  \[\int_{{}}^{{}}{\frac{x\ dx}{1-x\cot x}}=\]
  A)
             \[\log (\cos x-x\sin x)+c\] done clear
  B)
             \[\log (x\sin x-\cos x)+c\] done clear
  C)
             \[\log (\sin x-x\cos x)+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer9)
  \[\int_{{}}^{{}}{\frac{\sin 2x}{1+{{\sin }^{2}}x}dx=}\]     [Roorkee 1976]
  A)
             \[\log \sin 2x+c\] done clear
  B)
             \[\log (1+{{\sin }^{2}}x)+c\] done clear
  C)
             \[\frac{1}{2}\log (1+{{\sin }^{2}}x)+c\] done clear
  D)
             \[{{\tan }^{-1}}(\sin x)+c\] done clear
  View Solution play_arrow

• question_answer10)
  \[\int_{{}}^{{}}{\frac{{{x}^{3}}}{\sqrt{{{x}^{2}}+2}}dx=}\]
  A)
             \[\frac{1}{3}{{({{x}^{2}}+2)}^{3/2}}+2{{({{x}^{2}}+2)}^{1/2}}+c\] done clear
  B)
             \[\frac{1}{3}{{({{x}^{2}}+2)}^{3/2}}-2{{({{x}^{2}}+2)}^{1/2}}+c\] done clear
  C)
             \[\frac{1}{3}{{({{x}^{2}}+2)}^{3/2}}+{{({{x}^{2}}+2)}^{1/2}}+c\] done clear
  D)
             \[\frac{1}{3}{{({{x}^{2}}+2)}^{3/2}}-{{({{x}^{2}}+2)}^{1/2}}+c\] done clear
  View Solution play_arrow

• question_answer11)
  \[\int_{{}}^{{}}{\frac{{{x}^{e-1}}+{{e}^{x-1}}}{{{x}^{e}}+{{e}^{x}}}dx=}\]
  A)
             \[\log ({{x}^{e}}+{{e}^{x}})+c\] done clear
  B)
             \[e\log ({{x}^{e}}+{{e}^{x}})+c\] done clear
  C)
             \[\frac{1}{e}\log ({{x}^{e}}+{{e}^{x}})+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer12)
  \[\int_{{}}^{{}}{\frac{\sin x\ dx}{{{a}^{2}}+{{b}^{2}}{{\cos }^{2}}x}}=\]
  A)
             \[\log ({{a}^{2}}+{{b}^{2}}{{\cos }^{2}}x)+c\] done clear
  B)
             \[\frac{1}{ab}{{\tan }^{-1}}\left( \frac{a\cos x}{b} \right)+c\] done clear
  C)
             \[\frac{1}{ab}{{\cot }^{-1}}\left( \frac{b\cos x}{a} \right)+c\] done clear
  D)
             \[\frac{1}{ab}{{\cot }^{-1}}\left( \frac{a\cos x}{b} \right)+c\] done clear
  View Solution play_arrow

• question_answer13)
  \[\int_{{}}^{{}}{\sec x\log (\sec x+\tan x)\ dx=}\]
  A)
             \[{{[\log (\sec x+\tan x)]}^{2}}+c\]   done clear
  B)
             \[\frac{1}{2}{{[\log (\sec x+\tan x)]}^{2}}+c\] done clear
  C)
             \[{{\sec }^{2}}x+\tan x\sec x+c\]      done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer14)
  \[\int_{{}}^{{}}{\frac{x-2}{{{x}^{2}}-4x+3}dx=}\]               [MP PET 1987]
  A)
             \[\log \sqrt{{{x}^{2}}-4x+3}+c\] done clear
  B)
             \[x\log (x-3)-2\log (x-2)+c\] done clear
  C)
             \[\log [(x-3)(x-1)]\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer15)
  \[\int_{{}}^{{}}{\frac{3{{x}^{2}}}{{{x}^{6}}+1}dx=}\] [MNR 1981; MP PET 1988; RPET 1995]
  A)
             \[\log ({{x}^{6}}+1)+c\] done clear
  B)
             \[{{\tan }^{-1}}({{x}^{3}})+c\] done clear
  C)
             \[3{{\tan }^{-1}}({{x}^{3}})+c\] done clear
  D)
             \[3{{\tan }^{-1}}\left( \frac{{{x}^{3}}}{3} \right)+c\] done clear
  View Solution play_arrow

• question_answer16)
  \[\int_{{}}^{{}}{\frac{\cot x}{\log \sin x}}\ dx=\]                  [MNR 1974]
  A)
             \[\log (\log \sin x)+c\] done clear
  B)
             \[\log (\log \text{cosec}\,x)+c\] done clear
  C)
             \[2\log (\log \sin x)+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer17)
  \[\int_{{}}^{{}}{\frac{{{(1+\log x)}^{2}}}{x}}\ dx=\]         [Roorkee 1977]
  A)
             \[{{(1+\log x)}^{3}}+c\] done clear
  B)
             \[3{{(1+\log x)}^{3}}+c\] done clear
  C)
             \[\frac{1}{3}{{(1+\log x)}^{3}}+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer18)
  \[\int_{{}}^{{}}{{{\sec }^{p}}x\tan x\ dx=}\]
  A)
             \[\frac{{{\sec }^{p+1}}x}{p+1}+c\] done clear
  B)
             \[\frac{{{\sec }^{p}}x}{p}+c\] done clear
  C)
             \[\frac{{{\tan }^{p+1}}x}{p+1}+c\] done clear
  D)
             \[\frac{{{\tan }^{p}}x}{p}+c\] done clear
  View Solution play_arrow

• question_answer19)
  \[\int_{{}}^{{}}{\frac{dx}{{{e}^{x}}-1}=}\]                         [MP PET 1989]
  A)
             \[\ln (1-{{e}^{-x}})+c\] done clear
  B)
             \[-\ln (1-{{e}^{-x}})+c\] done clear
  C)
             \[\ln ({{e}^{x}}-1)+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer20)
  \[\int_{{}}^{{}}{{{x}^{2}}\sec {{x}^{3}}\ dx}=\] [MNR 1986; Roorkee 1975]
  A)
             \[\log (\sec {{x}^{3}}+\tan {{x}^{3}})\] done clear
  B)
             \[3(\sec {{x}^{3}}+\tan {{x}^{3}})\] done clear
  C)
             \[\frac{1}{3}\log (\sec {{x}^{3}}+\tan {{x}^{3}})\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer21)
  \[\int_{{}}^{{}}{\frac{\sin 2x}{{{\sin }^{4}}x+{{\cos }^{4}}x}dx=}\] [RPET 1995]
  A)
             \[{{\cot }^{-1}}({{\tan }^{2}}x)+c\] done clear
  B)
             \[{{\tan }^{-1}}({{\tan }^{2}}x)+c\] done clear
  C)
             \[{{\cot }^{-1}}({{\cot }^{2}}x)+c\] done clear
  D)
             \[{{\tan }^{-1}}({{\cot }^{2}}x)+c\] done clear
  View Solution play_arrow

• question_answer22)
  \[\int_{{}}^{{}}{\frac{x-2}{x(2\log x-x)}dx}=\]
  A)
             \[\log (2\log x-x)+c\] done clear
  B)
             \[\log \left( \frac{1}{2\log x-x} \right)+c\] done clear
  C)
             \[\log (x-2\log x)+c\] done clear
  D)
             \[\log \left( \frac{1}{x-2\log x} \right)+c\] done clear
  View Solution play_arrow

• question_answer23)
  \[\int_{{}}^{{}}{x\sqrt{1+{{x}^{2}}}}\ dx=\]                       [MP PET 1989]
  A)
             \[\frac{1+2{{x}^{2}}}{\sqrt{1+{{x}^{2}}}}+c\] done clear
  B)
             \[\sqrt{1+{{x}^{2}}}+c\] done clear
  C)
             \[3{{(1+{{x}^{2}})}^{3/2}}+c\] done clear
  D)
             \[\frac{1}{3}{{(1+{{x}^{2}})}^{3/2}}+c\] done clear
  View Solution play_arrow

• question_answer24)
  \[\int_{{}}^{{}}{\frac{{{e}^{x}}(x+1)}{{{\cos }^{2}}(x{{e}^{x}})}dx=}\] [Roorkee 1979; MP PET 1995; Pb. CET 2001]
  A)
             \[\tan (x{{e}^{x}})+c\] done clear
  B)
             \[\sec (x{{e}^{x}})\tan (x{{e}^{x}})+c\] done clear
  C)
             \[-\tan (x{{e}^{x}})+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer25)
  \[\int_{{}}^{{}}{\frac{\cos \sqrt{x}}{\sqrt{x}}}dx=\] [MP PET 1987; IIT 1990; SCRA 1996; RPET 2001]
  A)
             \[2\cos \sqrt{x}+c\] done clear
  B)
             \[2\sin \sqrt{x}+c\] done clear
  C)
             \[\sin \sqrt{x}+c\] done clear
  D)
             \[\frac{1}{2}\cos \sqrt{x}+c\] done clear
  View Solution play_arrow

• question_answer26)
  \[\int_{{}}^{{}}{\frac{x+1}{\sqrt{1+{{x}^{2}}}}dx}=\]        [MP PET 1991]
  A)
             \[\sqrt{1+{{x}^{2}}}+{{\tan }^{-1}}x+c\] done clear
  B)
             \[\sqrt{1+{{x}^{2}}}-\log \{x+\sqrt{1+{{x}^{2}}}\}+c\] done clear
  C)
             \[\sqrt{1+{{x}^{2}}}+\log \{x+\sqrt{1+{{x}^{2}}}\}+c\] done clear
  D)
             \[\sqrt{1+{{x}^{2}}}+\log (\sec x+\tan x)+c\] done clear
  View Solution play_arrow

• question_answer27)
  \[\int_{{}}^{{}}{\frac{\sin x\cos x}{a{{\cos }^{2}}x+b{{\sin }^{2}}x}dx=}\]            [AI CBSE 1988, 89]
  A)
             \[\frac{1}{2(b-a)}\log (a{{\cos }^{2}}x+b{{\sin }^{2}}x)+c\] done clear
  B)
             \[\frac{1}{b-a}\log (a{{\cos }^{2}}x+b{{\sin }^{2}}x)+c\] done clear
  C)
             \[\frac{1}{2}\log (a{{\cos }^{2}}x+b{{\sin }^{2}}x)+c\]                 done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer28)
  \[\int_{{}}^{{}}{\frac{{{e}^{{{\tan }^{-1}}x}}}{1+{{x}^{2}}}dx=}\] [MP PET 1987]
  A)
             \[\log (1+{{x}^{2}})+c\] done clear
  B)
             \[\log {{e}^{{{\tan }^{-1}}x}}+c\] done clear
  C)
             \[{{e}^{{{\tan }^{-1}}x}}+c\] done clear
  D)
             \[{{\tan }^{-1}}{{e}^{{{\tan }^{-1}}x}}+c\] done clear
  View Solution play_arrow

• question_answer29)
  \[\int_{{}}^{{}}{\frac{1}{x{{(\log x)}^{2}}}}\ dx=\]
  A)
             \[\frac{1}{\log x}+c\] done clear
  B)
             \[-\frac{1}{\log x}+c\] done clear
  C)
             \[\log \log x+c\] done clear
  D)
             \[-\log \log x+c\] done clear
  View Solution play_arrow

• question_answer30)
  \[\int_{{}}^{{}}{\frac{1}{\sqrt{x}}{{\tan }^{4}}\sqrt{x}}{{\sec }^{2}}\sqrt{x}\ dx=\]
  A)
             \[2{{\tan }^{5}}\sqrt{x}+c\] done clear
  B)
             \[\frac{1}{5}{{\tan }^{5}}\sqrt{x}+c\] done clear
  C)
             \[\frac{2}{5}{{\tan }^{5}}\sqrt{x}+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer31)
  \[\int_{{}}^{{}}{\frac{{{a}^{x}}}{\sqrt{1-{{a}^{2x}}}}dx=}\] [MNR 1983, 87]
  A)
             \[\frac{1}{\log a}{{\sin }^{-1}}{{a}^{x}}+c\] done clear
  B)
             \[{{\sin }^{-1}}{{a}^{x}}+c\] done clear
  C)
             \[\frac{1}{\log a}{{\cos }^{-1}}{{a}^{x}}+c\] done clear
  D)
             \[{{\cos }^{-1}}{{a}^{x}}+c\] done clear
  View Solution play_arrow

• question_answer32)
  \[\int_{{}}^{{}}{\frac{\sqrt{\tan x}}{\sin x\cos x}}\ dx=\] [Bihar CEE 1974; MP PET 2002; Kerala (Engg.) 2002]
  A)
             \[2\sqrt{\sec x}+c\] done clear
  B)
             \[2\sqrt{\tan x}+c\] done clear
  C)
             \[\frac{2}{\sqrt{\tan x}}+c\] done clear
  D)
             \[\frac{2}{\sqrt{\sec x}}+c\] done clear
  View Solution play_arrow

• question_answer33)
  \[\int_{{}}^{{}}{\frac{\sin 2x}{{{a}^{2}}+{{b}^{2}}{{\sin }^{2}}x}}\ dx=\] [Roorkee 1977]
  A)
             \[\frac{1}{{{b}^{2}}}\log ({{a}^{2}}+{{b}^{2}}{{\sin }^{2}}x)+c\]  done clear
  B)
             \[\frac{1}{b}\log ({{a}^{2}}+{{b}^{2}}{{\sin }^{2}}x)+c\] done clear
  C)
             \[\log ({{a}^{2}}+{{b}^{2}}{{\sin }^{2}}x)+c\]                                done clear
  D)
             \[{{b}^{2}}\log ({{a}^{2}}+{{b}^{2}}{{\sin }^{2}}x)+c\] done clear
  View Solution play_arrow

• question_answer34)
  \[\int_{{}}^{{}}{\frac{1}{x\sqrt{1+\log x}}\ dx=}\]           [Roorkee 1977]
  A)
             \[\frac{2}{3}{{(1+\log x)}^{3/2}}+c\] done clear
  B)
             \[{{(1+\log x)}^{3/2}}+c\] done clear
  C)
             \[2\sqrt{1+\log x}+c\] done clear
  D)
             \[\sqrt{1+\log x}+c\] done clear
  View Solution play_arrow

• question_answer35)
  \[\int_{{}}^{{}}{\frac{{{\sec }^{2}}x}{1+\tan x}\ dx=}\]    [MP PET 1987]
  A)
             \[\log (\cos x+\sin x)+c\] done clear
  B)
             \[\log ({{\sec }^{2}}x)+c\] done clear
  C)
             \[\log (1+\tan x)+c\] done clear
  D)
             \[-\frac{1}{{{(1+\tan x)}^{2}}}+c\] done clear
  View Solution play_arrow

• question_answer36)
  \[\int_{{}}^{{}}{\frac{{{e}^{2x}}-1}{{{e}^{2x}}+1}}\ dx=\] [MP PET 1987]
  A)
             \[\frac{{{e}^{2x}}-1}{{{e}^{2x}}+1}+c\] done clear
  B)
             \[\log ({{e}^{2x}}+1)-x+c\] done clear
  C)
             \[\log ({{e}^{2x}}+1)+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer37)
  \[\int_{{}}^{{}}{\frac{\cos \text{ec}x}{\log \tan \frac{x}{2}}\ dx=}\]
  A)
             \[\log \left( \log \tan \frac{x}{2} \right)+c\] done clear
  B)
             \[2\log \left( \log \tan \frac{x}{2} \right)+c\] done clear
  C)
             \[\frac{1}{2}\log \left( \log \tan \frac{x}{2} \right)+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer38)
  \[\int_{{}}^{{}}{\frac{1}{{{\cos }^{2}}x{{(1-\tan x)}^{2}}}dx=}\]
  A)
             \[\frac{1}{\tan x-1}+c\] done clear
  B)
             \[\frac{1}{1-\tan x}+c\] done clear
  C)
             \[-\frac{1}{3}\frac{1}{{{(1-\tan x)}^{3}}}+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer39)
  \[\int_{{}}^{{}}{\frac{10{{x}^{9}}+{{10}^{x}}{{\log }_{e}}10}{{{10}^{x}}+{{x}^{10}}}}\ dx=\] [MNR 1979]
  A)
             \[-\frac{1}{2}\frac{1}{{{({{10}^{x}}+{{x}^{10}})}^{2}}}+c\] done clear
  B)
             \[\log ({{10}^{x}}+{{x}^{10}})+c\] done clear
  C)
             \[\frac{1}{2}\frac{1}{{{({{10}^{x}}+{{x}^{10}})}^{2}}}+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer40)
  \[\int_{{}}^{{}}{\frac{1}{{{({{e}^{x}}+{{e}^{-x}})}^{2}}}\ dx=}\]
  A)
             \[-\frac{1}{2({{e}^{2x}}+1)}+c\] done clear
  B)
             \[\frac{1}{2({{e}^{2x}}+1)}+c\] done clear
  C)
             \[-\frac{1}{{{e}^{2x}}+1}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer41)
  \[\int_{{}}^{{}}{\frac{\cos 2x}{{{(\cos x+\sin x)}^{2}}}\ dx=}\]
  A)
             \[\log \sqrt{\cos x+\sin x}+c\] done clear
  B)
             \[\log (\cos x-\sin x)+c\] done clear
  C)
             \[\log (\cos x+\sin x)+c\] done clear
  D)
             \[-\frac{1}{\cos x+\sin x}+c\] done clear
  View Solution play_arrow

• question_answer42)
  \[\int_{{}}^{{}}{\frac{\tan (\log x)}{x}\ dx=}\]
  A)
             \[\log \cos (\log x)+c\] done clear
  B)
             \[\log \sin (\log x)+c\] done clear
  C)
             \[\log \sec (\log x)+c\] done clear
  D)
             \[\log \text{cosec}(\log x)+c\] done clear
  View Solution play_arrow

• question_answer43)
  \[\int_{{}}^{{}}{{{\cos }^{3}}x\ {{e}^{\log (\sin x)}}}\ dx\] is equal to
  A)
             \[-\frac{{{\sin }^{4}}x}{4}+c\] done clear
  B)
             \[-\frac{{{\cos }^{4}}x}{4}+c\] done clear
  C)
             \[\frac{{{e}^{\sin x}}}{4}+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer44)
  \[\int_{{}}^{{}}{\tan (3x-5)\sec (3x-5)\ dx=}\]                    [MP PET 1988]
  A)
             \[\sec (3x-5)+c\] done clear
  B)
             \[\frac{1}{3}\sec (3x-5)+c\] done clear
  C)
             \[\tan (3x-5)+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer45)
  \[\int_{{}}^{{}}{\frac{x}{1+{{x}^{4}}}\ dx=}\]      [IIT 1978; UPSEAT 2002]
  A)
             \[\frac{1}{2}{{\cot }^{-1}}{{x}^{2}}+c\] done clear
  B)
             \[\frac{1}{2}{{\tan }^{-1}}{{x}^{2}}+c\] done clear
  C)
             \[{{\cot }^{-1}}{{x}^{2}}+c\] done clear
  D)
             \[{{\tan }^{-1}}{{x}^{2}}+c\] done clear
  View Solution play_arrow

• question_answer46)
  \[\int_{{}}^{{}}{\frac{{{e}^{-x}}}{1+{{e}^{x}}}\ dx=}\]
  A)
             \[\log (1+{{e}^{x}})-x-{{e}^{-x}}+c\] done clear
  B)
             \[\log (1+{{e}^{x}})+x-{{e}^{-x}}+c\] done clear
  C)
             \[\log (1+{{e}^{x}})-x+{{e}^{-x}}+c\] done clear
  D)
             \[\log (1+{{e}^{x}})+x+{{e}^{-x}}+c\] done clear
  View Solution play_arrow

• question_answer47)
  \[\int_{{}}^{{}}{\frac{1}{\sqrt{1-{{e}^{2x}}}}\ dx=}\] [MP PET 1993, 2002; RPET 1999]
  A)
             \[x-\log [1+\sqrt{1-{{e}^{2x}}}]+c\] done clear
  B)
             \[x+\log [1+\sqrt{1-{{e}^{2x}}}]+c\] done clear
  C)
             \[\log [1+\sqrt{1-{{e}^{2x}}}]-x+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer48)
  \[\int_{{}}^{{}}{\frac{3{{x}^{2}}}{\sqrt{9-16{{x}^{6}}}}}\ dx=\]
  A)
             \[\frac{1}{4}{{\sin }^{-1}}\left( \frac{4{{x}^{3}}}{3} \right)+c\] done clear
  B)
             \[\frac{1}{3}{{\sin }^{-1}}\left( \frac{4{{x}^{3}}}{3} \right)+c\] done clear
  C)
             \[\frac{1}{4}{{\sin }^{-1}}{{x}^{3}}+c\] done clear
  D)
             \[\frac{1}{3}{{\sin }^{-1}}{{x}^{3}}+c\] done clear
  View Solution play_arrow

• question_answer49)
  \[\int_{{}}^{{}}{\cos x\sqrt{4-{{\sin }^{2}}x}}\ dx=\]
  A)
             \[\frac{1}{2}\sin x\sqrt{4-{{\sin }^{2}}x}-2{{\sin }^{-1}}\left( \frac{1}{2}\sin x \right)+c\] done clear
  B)
             \[\frac{1}{2}\sin x\sqrt{4-{{\sin }^{2}}x}+2{{\sin }^{-1}}\left( \frac{1}{2}\sin x \right)+c\] done clear
  C)
             \[\frac{1}{2}\sin x\sqrt{4-{{\sin }^{2}}x}+{{\sin }^{-1}}\left( \frac{1}{2}\sin x \right)+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer50)
  \[\int_{{}}^{{}}{{{x}^{2}}{{(3)}^{{{x}^{3}}+1}}dx=}\]
  A)
             \[{{(3)}^{{{x}^{3}}}}+c\] done clear
  B)
             \[\frac{{{(3)}^{{{x}^{3}}}}}{\log 3}+c\] done clear
  C)
             \[\log 3{{(3)}^{{{x}^{3}}}}+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer51)
  \[\int_{{}}^{{}}{{{\sec }^{2/3}}x\,\text{cose}{{\text{c}}^{4/3}}x\ dx=}\]
  A)
             \[-3{{(\tan x)}^{1/3}}+c\] done clear
  B)
             \[-3{{(\tan x)}^{-1/3}}+c\] done clear
  C)
             \[3{{(\tan x)}^{-1/3}}+c\] done clear
  D)
             \[{{(\tan x)}^{-1/3}}+c\] done clear
  View Solution play_arrow

• question_answer52)
  \[\int_{{}}^{{}}{{{\cos }^{5}}x\ dx=}\]
  A)
             \[\sin x-\frac{2}{3}{{\sin }^{3}}x+\frac{1}{5}{{\sin }^{5}}x+c\] done clear
  B)
             \[\sin x+\frac{2}{3}{{\sin }^{3}}x+\frac{1}{5}{{\sin }^{5}}x+c\] done clear
  C)
             \[\sin x-\frac{2}{3}{{\sin }^{3}}x-\frac{1}{5}{{\sin }^{5}}x+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer53)
  \[\int_{{}}^{{}}{\sec x{{\tan }^{3}}x\ dx=}\]
  A)
             \[\frac{1}{3}{{\sec }^{3}}x-\sec x+c\] done clear
  B)
             \[{{\sec }^{3}}x-\sec x+c\] done clear
  C)
             \[\frac{1}{3}{{\sec }^{3}}x+\sec x+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer54)
  \[\int_{{}}^{{}}{\frac{d\theta }{\sin \theta {{\cos }^{3}}\theta }=}\]
  A)
             \[\log \tan \theta +{{\tan }^{2}}\theta +c\] done clear
  B)
             \[\log \tan \theta -\frac{1}{2}{{\tan }^{2}}\theta +c\] done clear
  C)
             \[\log \tan \theta +\frac{1}{2}{{\tan }^{2}}\theta +c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer55)
  \[\int_{{}}^{{}}{\frac{1}{{{\cos }^{-1}}x.\sqrt{1-{{x}^{2}}}}dx=}\]
  A)
             \[\log ({{\cos }^{-1}}x)+c\] done clear
  B)
             \[-\log ({{\cos }^{-1}}x)+c\] done clear
  C)
             \[-\frac{1}{2{{({{\cos }^{-1}}x)}^{2}}}+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer56)
  To evaluate \[\int_{{}}^{{}}{{{x}^{3}}{{e}^{3{{x}^{2}}+5}}}dx\], the simplest way is to  
  A)
             Substitute \[{{x}^{2}}=t\] done clear
  B)
             Substitute \[(3{{x}^{2}}+5)=t\] done clear
  C)
             Integrate by parts done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer57)
  To evaluate \[\int_{{}}^{{}}{\frac{{{\sec }^{2}}x}{(1+\tan x)(2+\tan x)}\ dx}\], the most suitable substitution is  
  A)
             \[1+\tan x=t\] done clear
  B)
             \[2+\tan x=t\] done clear
  C)
             \[\tan x=t\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer58)
  \[\int_{{}}^{{}}{\frac{\text{cose}{{\text{c}}^{2}}x}{1+\cot x}dx=}\] [MNR 1973]
  A)
             \[\log (1+\cot x)+c\] done clear
  B)
             \[-\log (1+\cot x)+c\] done clear
  C)
             \[\frac{1}{2{{(1+\cot x)}^{2}}}+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer59)
  \[\int_{{}}^{{}}{\frac{1}{\sqrt{x}}}\sin \sqrt{x}\ dx=\]       [MP PET 1989]
  A)
             \[-\frac{1}{2}\cos \sqrt{x}+c\] done clear
  B)
             \[-2\cos \sqrt{x}+c\] done clear
  C)
             \[\frac{1}{2}\cos \sqrt{x}+c\] done clear
  D)
             \[2\cos \sqrt{x}+c\] done clear
  View Solution play_arrow

• question_answer60)
  \[\int_{{}}^{{}}{{{e}^{x}}{{\tan }^{2}}({{e}^{x}})dx=}\]
  A)
             \[\tan ({{e}^{x}})-x+c\] done clear
  B)
             \[{{e}^{x}}(\tan {{e}^{x}}-1)+c\] done clear
  C)
             \[\sec ({{e}^{x}})+c\] done clear
  D)
             \[\tan ({{e}^{x}})-{{e}^{x}}+c\] done clear
  View Solution play_arrow

• question_answer61)
  \[\int_{{}}^{{}}{\frac{dx}{{{e}^{-2x}}{{({{e}^{2x}}+1)}^{2}}}=}\]
  A)
             \[\frac{-1}{2({{e}^{2x}}+1)}+c\] done clear
  B)
             \[\frac{1}{2({{e}^{2x}}+1)}+c\] done clear
  C)
             \[\frac{1}{{{e}^{2x}}+1}+c\] done clear
  D)
             \[\frac{-1}{{{e}^{2x}}+1}+c\] done clear
  View Solution play_arrow

• question_answer62)
  . \[\int_{{}}^{{}}{{{\tan }^{4}}x\ dx=}\]
  A)
             \[{{\tan }^{3}}x-\tan x+x+c\] done clear
  B)
             \[\frac{1}{3}{{\tan }^{3}}x-\tan x+x+c\] done clear
  C)
             \[\frac{1}{3}{{\tan }^{3}}x+\tan x+x+c\] done clear
  D)
             \[\frac{1}{3}{{\tan }^{3}}x+\tan x+2x+c\] done clear
  View Solution play_arrow

• question_answer63)
  \[\int_{{}}^{{}}{\frac{dx}{x\sqrt{1-{{(\log x)}^{2}}}}=}\]
  A)
             \[{{\cos }^{-1}}(\log x)+c\] done clear
  B)
             \[x\log (1-{{x}^{2}})+c\] done clear
  C)
             \[{{\sin }^{-1}}(\log x)+c\] done clear
  D)
             \[\frac{1}{2}{{\cos }^{-1}}(\log x)+c\] done clear
  View Solution play_arrow

• question_answer64)
  \[\int_{{}}^{{}}{\frac{f'(x)}{{{[f(x)]}^{2}}}}\ dx=\]
  A)
             \[-{{[f(x)]}^{-1}}+c\] done clear
  B)
             \[\log [f(x)]+c\] done clear
  C)
             \[{{e}^{f(x)}}+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer65)
  For which of the following functions, the substitution \[{{x}^{2}}=t\]is applicable   
  A)
             \[\int_{{}}^{{}}{{{x}^{6}}{{\tan }^{-1}}{{x}^{3}}}\ dx\] done clear
  B)
             \[\int_{{}}^{{}}{{{\tan }^{-1}}\left( \frac{2x}{1-{{x}^{2}}} \right)\ dx}\] done clear
  C)
             \[\int_{{}}^{{}}{{{x}^{3}}\cos {{x}^{2}}\ dx}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer66)
  \[\int_{{}}^{{}}{\tan x}{{\sec }^{2}}x\sqrt{1-{{\tan }^{2}}x}\ dx=\]
  A)
             \[-\frac{1}{3}{{(1-{{\tan }^{2}}x)}^{3/2}}+c\] done clear
  B)
             \[\frac{1}{3}{{(1-{{\tan }^{2}}x)}^{3/2}}+c\] done clear
  C)
             \[-\frac{2}{3}{{(1-{{\tan }^{2}}x)}^{2/3}}+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer67)
  \[\int_{{}}^{{}}{\frac{\sin 2x}{\sin 5x\sin 3x}}\ dx=\]
  A)
             \[\log \sin 3x-\log \sin 5x+c\]   done clear
  B)
             \[\frac{1}{3}\log \sin 3x+\frac{1}{5}\log \sin 5x+c\] done clear
  C)
             \[\frac{1}{3}\log \sin 3x-\frac{1}{5}\log \sin 5x+c\] done clear
  D)
             \[3\log \sin 3x-5\log \sin 5x+c\] done clear
  View Solution play_arrow

• question_answer68)
  \[\int_{{}}^{{}}{\frac{{{e}^{x}}\ dx}{\sqrt{1-{{e}^{2x}}}}=}\]
  A)
             \[{{\cos }^{-1}}({{e}^{x}})+c\] done clear
  B)
             \[-{{\cos }^{-1}}({{e}^{x}})+c\] done clear
  C)
             \[{{\cos }^{-1}}({{e}^{2x}})+c\] done clear
  D)
             \[\sqrt{1-{{e}^{2x}}}+c\] done clear
  View Solution play_arrow

• question_answer69)
  \[\int_{{}}^{{}}{\frac{1}{\log a}({{a}^{x}}\cos {{a}^{x}})dx=}\]
  A)
             \[\sin {{a}^{x}}+c\] done clear
  B)
             \[{{a}^{x}}\sin {{a}^{x}}+c\] done clear
  C)
             \[\frac{1}{{{(\log a)}^{2}}}\sin {{a}^{x}}+c\] done clear
  D)
             \[\log \sin {{a}^{x}}+c\] done clear
  View Solution play_arrow

• question_answer70)
  \[\int_{{}}^{{}}{\frac{\sin x\ dx}{{{(a+b\cos x)}^{2}}}=}\]
  A)
             \[\frac{1}{b}(a+b\cos x)+c\] done clear
  B)
             \[\frac{1}{b(a+b\cos x)}+c\] done clear
  C)
             \[\frac{1}{b}\log (a+b\cos x)+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer71)
  \[\int_{{}}^{{}}{\frac{1}{{{x}^{3}}}{{[\log {{x}^{x}}]}^{2}}\ dx=}\]
  A)
             \[\frac{{{x}^{3}}}{3}(\log x)+x+c\] done clear
  B)
             \[\frac{1}{3}{{(\log x)}^{3}}+c\] done clear
  C)
             \[3\log (\log x)+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer72)
  \[\int_{{}}^{{}}{\frac{1}{x}{{\sec }^{2}}(\log x)dx=}\]
  A)
             \[\tan (\log x)+c\] done clear
  B)
             \[\log (\sec x)+c\] done clear
  C)
             \[\log (\tan x)+c\] done clear
  D)
             \[\sec (\log x)\ .\ \tan (\log x)+c\] done clear
  View Solution play_arrow

• question_answer73)
  \[\int_{{}}^{{}}{\frac{dx}{x\log x\log (\log x)}=}\]
  A)
             \[2\log (\log x)+c\] done clear
  B)
             \[\log [\log (\log x)]+c\] done clear
  C)
             \[\log (x\log x)+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer74)
   \[\int_{{}}^{{}}{\frac{{{\sec }^{2}}x\ dx}{\sqrt{{{\tan }^{2}}x+4}}=}\]
  A)
             \[\log \left[ \tan x+\sqrt{{{\tan }^{2}}x+4} \right]+c\]                done clear
  B)
             \[\frac{1}{2}\log \left[ \tan x+\sqrt{{{\tan }^{2}}x+4} \right]+c\] done clear
  C)
             \[\log \left[ \frac{1}{2}\tan x+\frac{1}{2}\sqrt{{{\tan }^{2}}x+4} \right]+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer75)
  \[\int_{{}}^{{}}{\frac{2x{{\tan }^{-1}}{{x}^{2}}}{1+{{x}^{4}}}}\ dx=\] [Roorkee 1982]
  A)
             \[{{[{{\tan }^{-1}}{{x}^{2}}]}^{2}}+c\] done clear
  B)
             \[\frac{1}{2}{{[{{\tan }^{-1}}{{x}^{2}}]}^{2}}+c\] done clear
  C)
             \[2{{[{{\tan }^{-1}}{{x}^{2}}]}^{2}}+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer76)
  \[\int_{{}}^{{}}{\frac{{{a}^{\sqrt{x}}}}{\sqrt{x}}dx=}\] [Roorkee 1990; MP PET 2001]
  A)
             \[2{{a}^{\sqrt{x}}}{{\log }_{e}}a+c\] done clear
  B)
             \[2{{a}^{\sqrt{x}}}{{\log }_{a}}e+c\] done clear
  C)
             \[2{{a}^{\sqrt{x}}}{{\log }_{10}}a+c\] done clear
  D)
             \[2{{a}^{\sqrt{x}}}{{\log }_{a}}10+c\] done clear
  View Solution play_arrow

• question_answer77)
  \[\int_{{}}^{{}}{\frac{{{x}^{3}}}{\sqrt{1-{{x}^{8}}}}dx=}\]
  A)
             \[\frac{1}{2}{{\sin }^{-1}}({{x}^{4}})+c\] done clear
  B)
             \[\frac{1}{3}{{\sin }^{-1}}({{x}^{4}})+c\] done clear
  C)
             \[\frac{1}{4}{{\sin }^{-1}}({{x}^{4}})+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer78)
  \[\int_{{}}^{{}}{2x{{\cos }^{3}}{{x}^{2}}\sin {{x}^{2}}dx=}\]
  A)
             \[-\frac{1}{4}{{\cos }^{4}}{{x}^{2}}+c\] done clear
  B)
             \[\frac{1}{4}{{\cos }^{4}}{{x}^{2}}+c\] done clear
  C)
             \[{{\cos }^{4}}{{x}^{2}}+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer79)
  \[\int_{{}}^{{}}{{{\sec }^{4}}x\tan x\ dx=}\] [AI CBSE 1980, 81; SCRA 1996]
  A)
             \[\frac{1}{4}{{\sec }^{4}}x+c\] done clear
  B)
             \[4{{\sec }^{4}}x+c\] done clear
  C)
             \[\frac{{{\sec }^{3}}x}{3}+c\] done clear
  D)
             \[3{{\sec }^{3}}x+c\] done clear
  View Solution play_arrow

• question_answer80)
  \[\int_{{}}^{{}}{{{e}^{-x}}\text{cose}{{\text{c}}^{2}}(2{{e}^{-x}}+5)}\ dx=\]        [AISSE 1988]
  A)
             \[\frac{1}{2}\cot (2{{e}^{-x}}+5)+c\] done clear
  B)
             \[-\frac{1}{2}\cot (2{{e}^{-x}}+5)+c\] done clear
  C)
             \[2\cot (2{{e}^{-x}}+5)+c\] done clear
  D)
             \[-2\cot (2{{e}^{-x}}+5)+c\] done clear
  View Solution play_arrow

• question_answer81)
  \[\int_{{}}^{{}}{{{\sin }^{3}}x\ .\ \cos x\ dx=}\]                   [SCRA 1996]
  A)
             \[\frac{{{\sin }^{4}}x{{\cos }^{2}}x}{8}+c\] done clear
  B)
             \[\frac{{{\sin }^{4}}x}{4}+c\] done clear
  C)
             \[\frac{{{\sin }^{2}}x}{2}+c\] done clear
  D)
             \[4{{\sin }^{4}}x+c\] done clear
  View Solution play_arrow

• question_answer82)
  \[\int_{{}}^{{}}{{{a}^{3x+3}}dx}=\]                                  [Roorkee 1977]
  A)
             \[\frac{{{a}^{3x+3}}}{\log a}+c\] done clear
  B)
             \[\frac{{{a}^{3x+3}}}{3\log a}+c\] done clear
  C)
             \[{{a}^{3x+3}}\log a+c\] done clear
  D)
             \[3{{a}^{3x+3}}\log a+c\] done clear
  View Solution play_arrow

• question_answer83)
  \[\int_{{}}^{{}}{\frac{\cos 2x+x+1}{{{x}^{2}}+\sin 2x+2x}}\ dx=\] [AI CBSE 1980]
  A)
             \[\log ({{x}^{2}}+\sin 2x+2x)+c\]      done clear
  B)
             \[-\log ({{x}^{2}}+\sin 2x+2x)+c\] done clear
  C)
             \[\frac{1}{2}\log ({{x}^{2}}+\sin 2x+2x)+c\]                                 done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer84)
  \[\int_{{}}^{{}}{\frac{1+\tan x}{x+\log \sec x}\ dx=}\]    [AI CBSE 1986]
  A)
             \[\log (x+\log \sec x)+c\] done clear
  B)
             \[-\log (x+\log \sec x)+c\] done clear
  C)
             \[\log (x-\log \sec x)+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer85)
  \[\int_{{}}^{{}}{\frac{(x+1){{(x+\log x)}^{2}}}{x}dx=}\]   [AI CBSE 1986]
  A)
             \[\frac{1}{3}(x+\log x)+c\] done clear
  B)
             \[\frac{1}{3}{{(x+\log x)}^{2}}+c\] done clear
  C)
             \[\frac{1}{3}{{(x+\log x)}^{3}}+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer86)
  \[\int_{{}}^{{}}{\frac{1+{{x}^{2}}}{\sqrt{1-{{x}^{2}}}}dx=}\]    [IIT 1977]
  A)
             \[\frac{3}{2}{{\sin }^{-1}}x-\frac{1}{2}x\sqrt{1-{{x}^{2}}}+c\] done clear
  B)
             \[\frac{3}{2}{{\sin }^{-1}}x+\frac{1}{2}x\sqrt{1-{{x}^{2}}}+c\] done clear
  C)
             \[\frac{3}{2}{{\cos }^{-1}}x-\frac{1}{2}x\sqrt{1-{{x}^{2}}}+c\] done clear
  D)
             \[\frac{3}{2}{{\cos }^{-1}}x+\frac{1}{2}x\sqrt{1-{{x}^{2}}}+c\] done clear
  View Solution play_arrow

• question_answer87)
  \[\int_{{}}^{{}}{\frac{\cos x-\sin x}{1+\sin 2x}\ dx=}\]       [AISSE 1985]
  A)
             \[-\frac{1}{\cos x+\sin x}+c\] done clear
  B)
             \[\frac{1}{\cos x+\sin x}+c\] done clear
  C)
             \[\frac{1}{\cos x-\sin x}+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer88)
  \[\int_{{}}^{{}}{{{x}^{3}}\sqrt{3+5{{x}^{4}}}}\ dx=\]            [DSSE 1982]
  A)
             \[{{(3+5{{x}^{4}})}^{3/2}}+c\] done clear
  B)
             \[\frac{1}{5}{{(3+5{{x}^{4}})}^{3/2}}+c\] done clear
  C)
             \[\frac{1}{30}{{(3+5{{x}^{4}})}^{3/2}}+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer89)
  \[\int_{{}}^{{}}{\sqrt{\frac{x}{{{a}^{3}}-{{x}^{3}}}}\ dx=}\]
  A)
             \[{{\sin }^{-1}}{{\left( \frac{x}{a} \right)}^{3/2}}+c\] done clear
  B)
             \[\frac{2}{3}{{\sin }^{-1}}{{\left( \frac{x}{a} \right)}^{3/2}}+c\] done clear
  C)
             \[\frac{3}{2}{{\sin }^{-1}}{{\left( \frac{x}{a} \right)}^{3/2}}+c\] done clear
  D)
             \[\frac{3}{2}{{\sin }^{-1}}{{\left( \frac{x}{a} \right)}^{2/3}}+c\] done clear
  View Solution play_arrow

• question_answer90)
  \[\int_{{}}^{{}}{\frac{1}{x{{\cos }^{2}}(1+\log x)}\ dx=}\]
  A)
             \[\tan \,(1+\log x)+c\] done clear
  B)
             \[\cot \,(1+\log x)+c\] done clear
  C)
             \[-\tan \,(1+\log x)+c\] done clear
  D)
             \[-\cot (\,1+\log x)+c\] done clear
  View Solution play_arrow

• question_answer91)
  \[\int_{{}}^{{}}{\frac{1}{{{x}^{2}}\sqrt{1+{{x}^{2}}}}}\ dx=\]
  A)
             \[-\frac{\sqrt{1+{{x}^{2}}}}{x}+c\] done clear
  B)
             \[\frac{\sqrt{1+{{x}^{2}}}}{x}+c\] done clear
  C)
             \[-\frac{\sqrt{1-{{x}^{2}}}}{x}+c\] done clear
  D)
             \[-\frac{\sqrt{{{x}^{2}}-1}}{x}+c\] done clear
  View Solution play_arrow

• question_answer92)
  \[\int_{{}}^{{}}{\frac{1}{({{x}^{2}}-1)\sqrt{{{x}^{2}}+1}}}\ dx=\]
  A)
             \[\frac{1}{2\sqrt{2}}\log \left\{ \frac{\sqrt{1+{{x}^{2}}}+x\sqrt{2}}{\sqrt{1+{{x}^{2}}}-x\sqrt{2}} \right\}+c\] done clear
  B)
             \[\frac{1}{2\sqrt{2}}\log \left\{ \frac{\sqrt{1+{{x}^{2}}}-\sqrt{2}}{\sqrt{1+{{x}^{2}}}+\sqrt{2}} \right\}+c\] done clear
  C)
             \[\frac{1}{2\sqrt{2}}\log \left\{ \frac{\sqrt{1+{{x}^{2}}}-x\sqrt{2}}{\sqrt{1+{{x}^{2}}}+x\sqrt{2}} \right\}+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer93)
  \[\int_{{}}^{{}}{\frac{\log (x+\sqrt{1+{{x}^{2}}})}{\sqrt{1+{{x}^{2}}}}\ dx=}\]
  A)
             \[\frac{1}{2}{{[\log (x+\sqrt{1+{{x}^{2}}})]}^{2}}+c\] done clear
  B)
             \[\log {{(x+\sqrt{1+{{x}^{2}}})}^{2}}+c\] done clear
  C)
             \[\log (x+\sqrt{1+{{x}^{2}}})+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer94)
  \[\int_{{}}^{{}}{{{e}^{x}}\sin ({{e}^{x}})}\ dx=\] [MP PET 1995]
  A)
             \[-\cos {{e}^{x}}+c\] done clear
  B)
             \[\cos {{e}^{x}}+c\] done clear
  C)
             \[-\text{cosec}\,{{e}^{x}}+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer95)
  \[\int_{{}}^{{}}{\frac{{{x}^{5}}\ dx}{\sqrt{(1+{{x}^{3}})}}=}\]        [IIT 1975]
  A)
             \[\frac{2}{3}\sqrt{(1+{{x}^{3}})}({{x}^{3}}+2)\] done clear
  B)
             \[\frac{2}{9}\sqrt{(1+{{x}^{3}})}({{x}^{3}}-4)\] done clear
  C)
             \[\frac{2}{9}\sqrt{(1+{{x}^{3}})}({{x}^{3}}+4)\] done clear
  D)
             \[\frac{2}{9}\sqrt{(1+{{x}^{3}})}({{x}^{3}}-2)\] done clear
  View Solution play_arrow

• question_answer96)
  \[\int_{{}}^{{}}{\frac{{{({{x}^{4}}-x)}^{1/4}}}{{{x}^{5}}}\ dx}\] is equal to
  A)
             \[\frac{4}{15}{{\left( 1-\frac{1}{{{x}^{3}}} \right)}^{5/4}}+c\] done clear
  B)
             \[\frac{4}{5}{{\left( 1-\frac{1}{{{x}^{3}}} \right)}^{5/4}}+c\] done clear
  C)
             \[\frac{4}{15}{{\left( 1+\frac{1}{{{x}^{3}}} \right)}^{5/4}}+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer97)
  \[\int_{{}}^{{}}{\frac{1}{{{[{{(x-1)}^{3}}{{(x+2)}^{5}}]}^{1/4}}}\ dx}\] is equal to
  A)
             \[\frac{4}{3}{{\left( \frac{x-1}{x+2} \right)}^{1/4}}+c\] done clear
  B)
             \[\frac{4}{3}{{\left( \frac{x+2}{x-1} \right)}^{1/4}}+c\] done clear
  C)
             \[\frac{1}{3}{{\left( \frac{x-1}{x+2} \right)}^{1/4}}+c\] done clear
  D)
             \[\frac{1}{3}{{\left( \frac{x+2}{x-1} \right)}^{1/4}}+c\] done clear
  View Solution play_arrow

• question_answer98)
  \[\int_{{}}^{{}}{\frac{1}{1+{{\sin }^{2}}x}\ dx=}\]
  A)
             \[\frac{1}{\sqrt{2}}{{\tan }^{-1}}(\sqrt{2}\tan x)+k\] done clear
  B)
             \[\sqrt{2}{{\tan }^{-1}}(\sqrt{2}\tan x)+k\] done clear
  C)
             \[-\frac{1}{\sqrt{2}}{{\tan }^{-1}}(\sqrt{2}\tan x)+k\] done clear
  D)
             \[-\sqrt{2}{{\tan }^{-1}}(\sqrt{2}\tan x)+k\] done clear
  View Solution play_arrow

• question_answer99)
  The value of \[\int_{{}}^{{}}{\frac{\sin x}{{{\cos }^{2}}x}\ dx}\] is
  A)
             \[\sin x+k\] done clear
  B)
             \[\tan x+k\] done clear
  C)
             \[\sec x+k\] done clear
  D)
             \[\tan x+\sec x+k\] done clear
  View Solution play_arrow

• question_answer100)
  The value of\[\int_{{}}^{{}}{{{e}^{x}}{{\sec }^{2}}({{e}^{x}})\ dx}\] is
  A)
             \[\tan ({{e}^{x}})+k\] done clear
  B)
             \[\tan ({{e}^{x}})\ .\ e+k\] done clear
  C)
             \[{{e}^{x}}\tan x+k\] done clear
  D)
             \[\frac{\tan ({{e}^{x}})}{{{e}^{x}}}+k\] done clear
  View Solution play_arrow

• question_answer101)
  The value of \[\int_{{}}^{{}}{\frac{dx}{x\sqrt{{{x}^{4}}-1}}}\] is
  A)
             \[\frac{1}{2}{{\sec }^{-1}}{{x}^{2}}+k\] done clear
  B)
             \[\log x\sqrt{{{x}^{4}}-1}+k\] done clear
  C)
             \[x\log \sqrt{{{x}^{4}}-1}+k\] done clear
  D)
             \[\log \sqrt{{{x}^{4}}-1}+k\] done clear
  View Solution play_arrow

• question_answer102)
  \[\int_{{}}^{{}}{\frac{t}{{{e}^{3{{t}^{2}}}}}\ dt=}\] [MP PET 1997]
  A)
             \[\frac{1}{6}{{e}^{3{{t}^{2}}}}+c\] done clear
  B)
             \[-\frac{1}{6}{{e}^{3{{t}^{2}}}}+c\] done clear
  C)
             \[\frac{1}{6}{{e}^{-3{{t}^{2}}}}+c\] done clear
  D)
             \[-\frac{1}{6}{{e}^{-3{{t}^{2}}}}+c\] done clear
  View Solution play_arrow

• question_answer103)
  If \[\int_{{}}^{{}}{\frac{1}{(1+x)\sqrt{x}}\ dx=f(x)+A}\], where A is any arbitrary constant, then the function \[f(x)\] is                                                                                             [MP PET 1998]
  A)
             \[2{{\tan }^{-1}}x\] done clear
  B)
             \[2{{\tan }^{-1}}\sqrt{x}\] done clear
  C)
             \[2{{\cot }^{-1}}\sqrt{x}\] done clear
  D)
             \[{{\log }_{e}}(1+x)\] done clear
  View Solution play_arrow

• question_answer104)
  \[\int_{{}}^{{}}{x\cos {{x}^{2}}\ dx}\] is equal to [MP PET 1999; Pb. CET 2000]
  A)
             \[-\frac{1}{2}{{\sin }^{2}}x+c\] done clear
  B)
             \[\frac{1}{2}{{\sin }^{2}}x+c\] done clear
  C)
             \[-\frac{1}{2}\sin {{x}^{2}}+c\] done clear
  D)
             \[\frac{1}{2}\sin {{x}^{2}}+c\] done clear
  View Solution play_arrow

• question_answer105)
  \[\int_{{}}^{{}}{\frac{{{x}^{2}}{{\tan }^{-1}}{{x}^{3}}}{1+{{x}^{6}}}\ dx}\] is equal to [MP PET 1999; UPSEAT 1999]
  A)
             \[{{\tan }^{-1}}({{x}^{3}})+c\] done clear
  B)
             \[\frac{1}{6}{{({{\tan }^{-1}}{{x}^{3}})}^{2}}+c\] done clear
  C)
             \[-\frac{1}{2}{{({{\tan }^{-1}}{{x}^{3}})}^{2}}+c\] done clear
  D)
             \[\frac{1}{2}{{({{\tan }^{-1}}{{x}^{2}})}^{3}}+c\] done clear
  View Solution play_arrow

• question_answer106)
  \[\int_{{}}^{{}}{\frac{{{x}^{2}}+1}{x({{x}^{2}}-1)}\ dx}\] is equal to            [MP PET 1999]
  A)
             \[\log \frac{{{x}^{2}}-1}{x}+c\] done clear
  B)
             \[-\log \frac{{{x}^{2}}-1}{x}+c\] done clear
  C)
             \[\log \frac{x}{{{x}^{2}}+1}+c\] done clear
  D)
             \[-\log \frac{x}{{{x}^{2}}+1}+c\] done clear
  View Solution play_arrow

• question_answer107)
  \[\int_{{}}^{{}}{\frac{{{e}^{2x}}+1}{{{e}^{2x}}-1}\ dx}\] equals    [RPET 1996]
  A)
             \[\log ({{e}^{x}}-{{e}^{-x}})+c\] done clear
  B)
             \[\log ({{e}^{x}}+{{e}^{-x}})+c\] done clear
  C)
             \[\log ({{e}^{-x}}-{{e}^{x}})+c\] done clear
  D)
             \[\log (1-{{e}^{-x}})+c\] done clear
  View Solution play_arrow

• question_answer108)
  \[\int_{{}}^{{}}{\frac{\cos x-\sin x}{\sqrt{\sin 2x}}\ dx}\] equals [RPET 1996]
  A)
             \[{{\cosh }^{-1}}(\sin x+\cos x)+c\]                                 done clear
  B)
             \[{{\sinh }^{-1}}(\sin x+\cos x)+c\] done clear
  C)
             \[-{{\cosh }^{-1}}(\sin x+\cos x)+c\] done clear
  D)
    \[-{{\sinh }^{-1}}(\sin x+\cos x)+c\] done clear
  View Solution play_arrow

• question_answer109)
  The value of \[\int_{{}}^{{}}{\left( 1+\frac{1}{{{x}^{2}}} \right)\ {{e}^{\left( x-\frac{1}{x} \right)}}}\ dx\] equals    [Kurukshetra CEE 1998]
  A)
             \[{{e}^{x-\frac{1}{x}}}+c\] done clear
  B)
             \[{{e}^{x+\frac{1}{x}}}+c\] done clear
  C)
             \[{{e}^{{{x}^{2}}-\frac{1}{x}}}+c\] done clear
  D)
             \[{{e}^{{{x}^{2}}+\frac{1}{{{x}^{2}}}}}+c\] done clear
  View Solution play_arrow

• question_answer110)
  \[\int_{{}}^{{}}{(x+3){{({{x}^{2}}+6x+10)}^{9}}\ dx}\] equals        [SCRA 1996]
  A)
             \[\frac{1}{20}{{({{x}^{2}}+6x+10)}^{10}}+c\]               done clear
  B)
             \[\frac{1}{20}{{(x+3)}^{2}}{{({{x}^{2}}+6x+10)}^{10}}+c\] done clear
  C)
             \[\frac{1}{16}{{({{x}^{2}}+6x+10)}^{8}}+c\]                 done clear
  D)
             \[\frac{1}{38}{{(x+3)}^{19}}+\frac{1}{2}(x+3)+c\] done clear
  View Solution play_arrow

• question_answer111)
  A primitive of \[\frac{x}{{{x}^{2}}+1}\] is              [SCRA 1996]
  A)
             \[{{\log }_{e}}({{x}^{2}}+1)\] done clear
  B)
             \[x{{\tan }^{-1}}x\] done clear
  C)
             \[\frac{{{\log }_{e}}({{x}^{2}}+1)}{2}\] done clear
  D)
             \[\frac{1}{2}x{{\tan }^{-1}}x\] done clear
  View Solution play_arrow

• question_answer112)
  \[\int_{{}}^{{}}{{{\sin }^{3}}x\ dx}\] is equal to [SCRA 1996]
  A)
             \[{{\sin }^{2}}x+1\] done clear
  B)
             \[\sin {{x}^{2}}+{{x}^{2}}+1\] done clear
  C)
             \[\frac{{{\cos }^{3}}x}{3}-\cos x\] done clear
  D)
             \[\frac{1}{4}{{\sin }^{4}}x-\frac{3}{4}{{\sin }^{2}}x\] done clear
  View Solution play_arrow

• question_answer113)
  \[\int_{{}}^{{}}{\frac{1}{x}\log x\ dx}\] is equal to [SCRA 1996]
  A)
             \[\frac{1}{2}\log x+c\] done clear
  B)
             \[\frac{1}{2}{{(\log x)}^{2}}+c\] done clear
  C)
             \[\frac{1}{2}\log {{(x)}^{2}}+c\] done clear
  D)
             \[\log x+c\] done clear
  View Solution play_arrow

• question_answer114)
  \[\int_{{}}^{{}}{{{\sin }^{2}}x\cos x\ dx}\] is equal to [SCRA 1996]
  A)
             \[\frac{{{\cos }^{2}}x}{2}+c\] done clear
  B)
             \[\frac{{{\sin }^{2}}x}{3}+c\] done clear
  C)
             \[\frac{{{\sin }^{3}}x}{3}+c\] done clear
  D)
             \[-\frac{{{\cos }^{2}}x}{2}+c\] done clear
  View Solution play_arrow

• question_answer115)
  \[\int_{{}}^{{}}{{{e}^{{{x}^{2}}}}x\ dx}\] is equal to [SCRA 1996]
  A)
             \[{{e}^{{{x}^{2}}}}\] done clear
  B)
             \[\frac{1}{2}{{e}^{{{x}^{2}}}}\] done clear
  C)
             \[2{{e}^{{{x}^{2}}}}\] done clear
  D)
             \[\frac{{{e}^{{{x}^{2}}}}-{{x}^{2}}}{2}\] done clear
  View Solution play_arrow

• question_answer116)
  The value of \[\int_{{}}^{{}}{\frac{{{x}^{3}}}{\sqrt{1+{{x}^{4}}}}\ dx}\] is                [SCRA 1996]
  A)
             \[{{(1+{{x}^{4}})}^{\frac{1}{2}}}+c\] done clear
  B)
             \[-{{(1+{{x}^{4}})}^{\frac{1}{2}}}+c\] done clear
  C)
             \[\frac{1}{2}{{(1+{{x}^{4}})}^{\frac{1}{2}}}+c\] done clear
  D)
             \[-\frac{1}{2}{{(1+{{x}^{4}})}^{\frac{1}{2}}}+c\] done clear
  View Solution play_arrow

• question_answer117)
  What is the value of the integral \[I=\int{\frac{dx}{(1+{{e}^{x}})\,\,(1+{{e}^{-x}})}}\] [DCE 1999]
  A)
             \[\frac{-1}{1+{{e}^{x}}}\] done clear
  B)
             \[\frac{{{e}^{x}}}{1+{{e}^{x}}}\] done clear
  C)
             \[\frac{1}{1+{{e}^{x}}}\] done clear
  D)
    None of these done clear
  View Solution play_arrow

• question_answer118)
  \[\int{\frac{{{e}^{\sqrt{x}}}}{\sqrt{x}}dx}=\]        [DCE 1999]
  A)
             \[{{e}^{\sqrt{x}}}\] done clear
  B)
             \[\frac{{{e}^{\sqrt{x}}}}{2}\] done clear
  C)
             \[2\,{{e}^{\sqrt{x}}}\] done clear
  D)
   \[\sqrt{x}\,.\,{{e}^{\sqrt{x}}}\] done clear
  View Solution play_arrow

• question_answer119)
  \[\int{\frac{{{\sin }^{3}}2x}{{{\cos }^{5}}2x}dx=}\] [Karnataka CET 1999]
  A)
             \[{{\tan }^{4}}x+C\] done clear
  B)
             \[\tan 4x+C\] done clear
  C)
             \[{{\tan }^{4}}2x+x+C\] done clear
  D)
    \[\frac{1}{8}{{\tan }^{4}}2x+C\] done clear
  View Solution play_arrow

• question_answer120)
  \[\int{{{x}^{x}}(1+\log x)\,\,dx}\] is equal to       [RPET 2000]
  A)
             \[{{x}^{x}}\] done clear
  B)
             \[{{x}^{2x}}\] done clear
  C)
             \[{{x}^{x}}\log x\] done clear
  D)
    \[\frac{1}{2}{{(1+\log x)}^{2}}\] done clear
  View Solution play_arrow

• question_answer121)
  \[\int{\frac{dx}{{{({{a}^{2}}+{{x}^{2}})}^{3/2}}}}\] is equal to       [RPET 2000]
  A)
             \[\frac{x}{{{\left( {{a}^{2}}+{{x}^{2}} \right)}^{1/2}}}\] done clear
  B)
             \[\frac{x}{{{a}^{2}}{{\left( {{a}^{2}}+{{x}^{2}} \right)}^{1/2}}}\] done clear
  C)
             \[\frac{1}{{{a}^{2}}{{\left( {{a}^{2}}+{{x}^{2}} \right)}^{1/2}}}\] done clear
  D)
    None of these done clear
  View Solution play_arrow

• question_answer122)
  \[\int_{{}}^{{}}{\frac{{{e}^{m{{\tan }^{-1}}x}}}{1+{{x}^{2}}}dx}\] equals to            [RPET 2001]
  A)
             \[{{e}^{{{\tan }^{-1}}x}}\] done clear
  B)
             \[\frac{1}{m}{{e}^{{{\tan }^{-1}}x}}\] done clear
  C)
             \[\frac{1}{m}{{e}^{m{{\tan }^{-1}}x}}\] done clear
  D)
    None of these done clear
  View Solution play_arrow

• question_answer123)
  \[\int{\frac{1+{{\tan }^{2}}x}{1-{{\tan }^{2}}x}\,dx}\] equals to   [RPET 2001]
  A)
             \[\log \left( \frac{1-\tan x}{1+\tan x} \right)+c\] done clear
  B)
             \[\log \left( \frac{1+\tan x}{1-\tan x} \right)+c\] done clear
  C)
             \[\frac{1}{2}\log \left( \frac{1-\tan x}{1+\tan x} \right)+c\] done clear
  D)
    \[\frac{1}{2}\log \left( \frac{1+\tan x}{1-\tan x} \right)+c\] done clear
  View Solution play_arrow

• question_answer124)
  The value of \[\int{\frac{2\,\,dx}{\sqrt{1-4{{x}^{2}}}}}\] is [Karnataka CET 2001; Pb. CET 2001]
  A)
             \[{{\tan }^{-1}}(2x)+c\] done clear
  B)
             \[{{\cot }^{-1}}(2x)+c\] done clear
  C)
             \[{{\cos }^{-1}}(2x)+c\] done clear
  D)
    \[{{\sin }^{-1}}(2x)+c\] done clear
  View Solution play_arrow

• question_answer125)
  \[\int{{{e}^{3\log x}}{{({{x}^{4}}+1)}^{-1}}\,\,dx}\]= [MP PET 2001]
  A)
             \[\log ({{x}^{4}}+1)+c\] done clear
  B)
             \[\frac{1}{4}\log ({{x}^{4}}+1)+c\] done clear
  C)
             \[-\log ({{x}^{4}}+1)+c\] done clear
  D)
    None of these done clear
  View Solution play_arrow

• question_answer126)
  \[\int_{{}}^{{}}{\frac{dx}{2\sqrt{x}(1+x)}=}\]      [RPET 2002]
  A)
             \[\frac{1}{2}{{\tan }^{-1}}(\sqrt{x})+c\] done clear
  B)
             \[{{\tan }^{-1}}(\sqrt{x})+c\] done clear
  C)
             \[2{{\tan }^{-1}}(\sqrt{x})+c\] done clear
  D)
    None of these done clear
  View Solution play_arrow

• question_answer127)
  \[\int{\text{cose}{{\text{c}}^{4}}x\,dx}=\]          [RPET 2002]
  A)
             \[\cot x+\frac{{{\cot }^{3}}x}{3}+c\] done clear
  B)
             \[\tan x+\frac{{{\tan }^{3}}x}{3}+c\] done clear
  C)
             \[-\cot x-\frac{{{\cot }^{3}}x}{3}+c\] done clear
  D)
    \[-\tan x-\frac{{{\tan }^{3}}x}{3}+c\] done clear
  View Solution play_arrow

• question_answer128)
  \[\int{x{{e}^{{{x}^{2}}}}}dx=\]                                  [RPET 2003]
  A)
             \[-\frac{{{e}^{{{x}^{2}}}}}{2}+c\] done clear
  B)
             \[\frac{{{e}^{{{x}^{2}}}}}{2}+c\] done clear
  C)
             \[\frac{{{e}^{x}}}{2}+c\] done clear
  D)
   \[-\frac{{{e}^{x}}}{2}+c\] done clear
  View Solution play_arrow

• question_answer129)
  . If \[\int{f(x)\,\,dx=g(x),}\] then \[\int{{{f}^{-1}}(x)}\,\,dx\] is equal to [MP PET 2003]
  A)
             \[{{g}^{-1}}(x)\] done clear
  B)
             \[x{{f}^{-1}}(x)-g({{f}^{-1}}(x))\] done clear
  C)
             \[x{{f}^{-1}}(x)-{{g}^{-1}}(x)\] done clear
  D)
             \[{{f}^{-1}}(x)\] done clear
  View Solution play_arrow

• question_answer130)
  The value of \[\int_{{}}^{{}}{\frac{{{e}^{x}}}{{{e}^{x}}+1}}\,dx\] is            [Pb. CET 2000]
  A)
             \[{{e}^{x}}+c\] done clear
  B)
             \[({{e}^{x}}+1)+c\] done clear
  C)
                     \[\log ({{e}^{x}}+1)+c\] done clear
  D)
   None of these done clear
  View Solution play_arrow

• question_answer131)
  The value of \[\int_{{}}^{{}}{\frac{\sin x-\cos x}{\sin x+\cos x}\,dx}\] is  [Pb. CET 2000]
  A)
             \[\frac{1}{\sin x+\cos x}+c\] done clear
  B)
             \[\frac{1}{\sin x-\cos x}+c\] done clear
  C)
                     \[\log (\sin x+\cos x)+c\] done clear
  D)
                     \[\log \left( \frac{1}{\sin x+\cos x} \right)+c\] done clear
  View Solution play_arrow

• question_answer132)
  \[\int_{{}}^{{}}{\frac{{{({{\tan }^{-1}}x)}^{3}}}{1+{{x}^{2}}}\,dx=}\]           [UPSEAT 2004]
  A)
             \[{{({{\tan }^{-1}}x)}^{4}}+c\] done clear
  B)
             \[\frac{{{({{\tan }^{-1}}x)}^{4}}}{4}+c\] done clear
  C)
                     \[2{{\tan }^{-1}}x+c\] done clear
  D)
   \[2{{({{\tan }^{-1}}x)}^{2}}+c\] done clear
  View Solution play_arrow

• question_answer133)
  \[\int_{{}}^{{}}{\sqrt{\frac{1-x}{1+x}}}\ dx=\]        [IIT 1971]
  A)
             \[{{\sin }^{-1}}x-\frac{1}{2}\sqrt{1-{{x}^{2}}}+c\] done clear
  B)
             \[{{\sin }^{-1}}x+\frac{1}{2}\sqrt{1-{{x}^{2}}}+c\] done clear
  C)
             \[{{\sin }^{-1}}x-\sqrt{1-{{x}^{2}}}+c\] done clear
  D)
             \[{{\sin }^{-1}}x+\sqrt{1-{{x}^{2}}}+c\] done clear
  View Solution play_arrow

• question_answer134)
  \[\int_{{}}^{{}}{\frac{\sqrt{x}}{1+x}dx=}\]
  A)
             \[\sqrt{x}-{{\tan }^{-1}}\sqrt{x}+c\] done clear
  B)
             \[2(\sqrt{x}-{{\tan }^{-1}}\sqrt{x})+c\] done clear
  C)
             \[2(\sqrt{x}+{{\tan }^{-1}}x)+c\] done clear
  D)
             \[\sqrt{1+x}+c\] done clear
  View Solution play_arrow

• question_answer135)
  \[\int_{{}}^{{}}{\frac{\sin x}{\sin x-\cos x}}\ dx=\]           [Roorkee 1988]
  A)
             \[\frac{1}{2}\log (\sin x-\cos x)+x+c\]                            done clear
  B)
             \[\frac{1}{2}[\log (\sin x-\cos x)+x]+c\] done clear
  C)
             \[\frac{1}{2}\log (\cos x-\sin x)+x+c\]                            done clear
  D)
             \[\frac{1}{2}[\log (\cos x-\sin x)+x]+c\] done clear
  View Solution play_arrow

• question_answer136)
  \[\int{\sqrt{\frac{1+x}{1-x}}\,\,dx=}\]                   [RPET 2002]
  A)
             \[-{{\sin }^{-1}}x-\sqrt{1-{{x}^{2}}}\,+c\] done clear
  B)
             \[{{\sin }^{-1}}x+\sqrt{1-{{x}^{2}}}\,+c\] done clear
  C)
             \[{{\sin }^{-1}}x-\sqrt{1-{{x}^{2}}}\,+c\] done clear
  D)
             \[-{{\sin }^{-1}}x-\sqrt{{{x}^{2}}-1}\,+c\] done clear
  View Solution play_arrow

• question_answer137)
  \[\int_{{}}^{{}}{\frac{x}{\sqrt{4-{{x}^{4}}}}dx}=\] [Roorkee 1976]
  A)
             \[{{\cos }^{-1}}\frac{{{x}^{2}}}{2}\] done clear
  B)
             \[\frac{1}{2}{{\cos }^{-1}}\frac{{{x}^{2}}}{2}\] done clear
  C)
             \[{{\sin }^{-1}}\frac{{{x}^{2}}}{2}\] done clear
  D)
             \[\frac{1}{2}{{\sin }^{-1}}\frac{{{x}^{2}}}{2}\] done clear
  View Solution play_arrow

• question_answer138)
  \[\int{\frac{\sin x\,\,dx}{3+4{{\cos }^{2}}x}=}\] [Karnataka CET 2000]
  A)
             \[\log (3+4{{\cos }^{2}}x)+c\] done clear
  B)
             \[\frac{-1}{2\sqrt{3}}{{\tan }^{-1}}\left( \frac{\cos x}{\sqrt{3}} \right)+c\]  done clear
  C)
             \[\frac{-1}{2\sqrt{3}}{{\tan }^{-1}}\left( \frac{2\cos x}{\sqrt{3}} \right)+c\] done clear
  D)
             \[\frac{1}{2\sqrt{3}}{{\tan }^{-1}}\left( \frac{2\cos x}{\sqrt{3}} \right)+c\] done clear
  View Solution play_arrow

• question_answer139)
  The value of \[\int_{{}}^{{}}{\frac{dx}{\sqrt{x}\,(x+9)}dx}\] is equal to    [Pb. CET 2002]
  A)
             \[{{\tan }^{-1}}\sqrt{x}\] done clear
  B)
             \[{{\tan }^{-1}}\left( \frac{\sqrt{x}}{3} \right)\] done clear
  C)
             \[\frac{2}{3}{{\tan }^{-1}}\sqrt{x}\] done clear
  D)
             \[\frac{2}{3}{{\tan }^{-1}}\left( \frac{\sqrt{x}}{3} \right)\] done clear
  View Solution play_arrow

• question_answer140)
  \[{{\int{\left\{ \frac{(\log x-1)}{1+{{(\log x)}^{2}}} \right\}}}^{2}}dx\] is equal to                 [AIEEE 2005]
  A)
                  \[\frac{x{{e}^{x}}}{1+{{x}^{2}}}+c\] done clear
  B)
                  \[\frac{x}{{{(\log x)}^{2}}+1}+C\] done clear
  C)
                  \[\frac{\log x}{{{(\log x)}^{2}}+1}+c\] done clear
  D)
                  \[\frac{x}{{{x}^{2}}+1}+c\] done clear
  View Solution play_arrow

• question_answer141)
  \[\int_{{}}^{{}}{\frac{\sin 2xdx}{1+{{\cos }^{2}}x}}=\]      [Karnataka CET 2005]
  A)
                  \[\frac{1}{2}\log (1+{{\cos }^{2}}x)+c\] done clear
  B)
                  \[2\log (1+{{\cos }^{2}}x)+c\] done clear
  C)
                  \[\frac{1}{2}\log (1+\cos 2x)+c\] done clear
  D)
                  \[-\log (1+{{\cos }^{2}}x)+c\] done clear
  View Solution play_arrow

• question_answer142)
  If \[\int{\frac{\cos 4x+1}{\cos x-\tan x}}dx=k\,\,\cos 4x+c\] then              [DCE 2005]
  A)
                  \[k=-1/2\] done clear
  B)
                  \[k=-1/8\] done clear
  C)
                  \[k=-1/4\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer143)
  If \[\int{\frac{1}{x+{{x}^{5}}}dx=f(x)+c}\], then the value of \[\int{\frac{{{x}^{4}}}{x+{{x}^{5}}}dx}\] is [DCE 2005]
  A)
                  \[\log x-f(x)+c\] done clear
  B)
                  \[f(x)+\log x+c\] done clear
  C)
                  \[f(x)-\log x+c\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer144)
  Let \[f(x)=\int{\frac{{{x}^{2}}dx}{(1+{{x}^{2}})\,\left( 1+\sqrt{1+{{x}^{2}}} \right)}}\]and \[f(0)=0\], then the value of \[f(1)\] be           [AMU 2005]
  A)
                  \[\log (1+\sqrt{2})\] done clear
  B)
                  \[\log (1+\sqrt{2})-\frac{\pi }{4}\] done clear
  C)
                  \[\log (1+\sqrt{2})+\frac{\pi }{2}\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer145)
  \[\int{\sqrt{{{e}^{x}}-1}}dx=\]   [Kerala (Engg.) 2005]
  A)
                  \[2\left[ \sqrt{{{e}^{x}}-1}-{{\tan }^{-1}}\sqrt{{{e}^{x}}-1} \right]+c\] done clear
  B)
                  \[\sqrt{{{e}^{x}}-1}-{{\tan }^{-1}}\sqrt{{{e}^{x}}-1}+c\] done clear
  C)
                  \[\sqrt{{{e}^{x}}-1}+{{\tan }^{-1}}\sqrt{{{e}^{x}}-1}+c\] done clear
  D)
                  \[2\left[ \sqrt{{{e}^{x}}-1}+{{\tan }^{-1}}\sqrt{{{e}^{x}}-1} \right]+c\] done clear
  E)
                  \[2\left[ \sqrt{{{e}^{x}}-1}-{{\tan }^{-1}}\sqrt{{{e}^{x}}+1} \right]+c\] done clear
  View Solution play_arrow

• question_answer146)
  \[\int{\frac{dx}{\sin (x-a)\sin (x-b)}}\] is                [Kerala (Engg.) 2005]
  A)
                  \[\frac{1}{\sin (a-b)}\log \left| \frac{\sin (x-a)}{\sin (x-b)} \right|+c\]     done clear
  B)
                  \[\frac{-1}{\sin (a-b)}\log \left| \frac{\sin (x-a)}{\sin (x-b)} \right|+c\] done clear
  C)
                  \[\log \sin (x-a)\sin (x-b)+c\] done clear
  D)
                  \[\log \left| \frac{\sin (x-a)}{\sin (x-b)} \right|\] done clear
  E)
                  \[\frac{1}{\sin (x-a)}\log \sin (x-a)\sin (x-b)+c\] done clear
  View Solution play_arrow

• question_answer147)
  \[\int{\frac{(\sin \theta +\cos \theta )}{\sqrt{\sin 2\theta }}}d\theta =\]                [Kerala (Engg.) 2005]
  A)
                  \[\log \left| \cos \theta -\sin \theta +\sqrt{\sin 2\theta } \right|\] done clear
  B)
                  \[\log \left| \sin \theta -\cos \theta )+\sqrt{\sin 2\theta } \right|\] done clear
  C)
                  \[{{\sin }^{-1}}(\sin \theta -\cos \theta )+c\]       done clear
  D)
                  \[{{\sin }^{-1}}(\sin \theta +\cos \theta )+c\] done clear
  E)
                  \[{{\sin }^{-1}}(\cos \theta -\sin \theta )+c\] done clear
  View Solution play_arrow

• question_answer148)
  \[\int{{{\cos }^{-3/7}}}x{{\sin }^{-11/7}}x\,\,dx=\]            [Kerala (Engg.) 2005]
  A)
                  \[\log |{{\sin }^{4/7}}x|+c\] done clear
  B)
                  \[\frac{4}{7}{{\tan }^{4/7}}x+c\] done clear
  C)
                  \[\frac{-7}{4}{{\tan }^{-4/7}}x+c\] done clear
  D)
                  \[\log |{{\cos }^{3/7}}x|+c\] done clear
  E)
                  \[\frac{7}{4}{{\tan }^{-4/7}}x+C\] done clear
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