JEE Main & Advanced Mathematics Definite Integration Question Bank

done Summation of series by Definite Integration, Gamma function, Leibnitz's rule Total Questions - 54

• question_answer1)
  The value of \[\underset{n\to \infty }{\mathop{\lim }}\,\left[ \frac{n}{1+{{n}^{2}}}+\frac{n}{4+{{n}^{2}}}+\frac{n}{9+{{n}^{2}}}+....+\frac{1}{2n} \right]\]is equal to                     [Bihar CEE 1994]
  A)
                  \[\frac{\pi }{2}\] done clear
  B)
                  \[\frac{\pi }{4}\] done clear
  C)
                  1              done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer2)
  \[\underset{n\to \infty }{\mathop{\lim }}\,\frac{1}{{{1}^{3}}+{{n}^{3}}}+\frac{4}{{{2}^{3}}+{{n}^{3}}}+....+\frac{1}{2n}\]is equal to           [RPET 1997]
  A)
                  \[\frac{1}{3}{{\log }_{e}}3\]         done clear
  B)
                  \[\frac{1}{3}{{\log }_{e}}2\] done clear
  C)
                  \[\frac{1}{3}{{\log }_{e}}\frac{1}{3}\]      done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer3)
  \[\underset{x\to a}{\mathop{\lim }}\,\frac{f(x)-f(a)}{g(x)\,-g(a)},\]         [EAMCET 1994]
  A)
                  \[\frac{9}{100}\]               done clear
  B)
                  \[-1/2\] done clear
  C)
                  \[\frac{1}{99}\] done clear
  D)
                  \[\frac{1}{101}\] done clear
  View Solution play_arrow

• question_answer4)
  \[\underset{n\to \infty }{\mathop{\lim }}\,{{\left[ \frac{n!}{{{n}^{n}}} \right]}^{1/n}}\]equals    [Kurukshetra CEE 1998]
  A)
                  e              done clear
  B)
                  \[1/e\] done clear
  C)
                  \[\pi /4\]              done clear
  D)
                  \[4/\pi \] done clear
  View Solution play_arrow

• question_answer5)
  \[\underset{n\to \infty }{\mathop{\lim }}\,\frac{1}{n}\sum\limits_{r=1}^{2n}{\frac{r}{\sqrt{{{n}^{2}}+{{r}^{2}}}}}\] equals   [IIT 1997 Re-exam]
  A)
                  \[1+\sqrt{5}\]    done clear
  B)
                  \[-1+\sqrt{5}\] done clear
  C)
                  \[-1+\sqrt{2}\]  done clear
  D)
                  \[1+\sqrt{2}\] done clear
  View Solution play_arrow

• question_answer6)
  \[\underset{n\to \infty }{\mathop{\lim }}\,\left[ \frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}+.....\frac{1}{2n} \right]=\]                  [Karnataka CET 1999]
  A)
                  0              done clear
  B)
                  \[{{\log }_{e}}4\] done clear
  C)
                  \[{{\log }_{e}}3\]              done clear
  D)
                  \[{{\log }_{e}}2\] done clear
  View Solution play_arrow

• question_answer7)
  \[\underset{n\to \infty }{\mathop{\lim }}\,\sum\limits_{k=1}^{n}{\frac{k}{{{n}^{2}}+{{k}^{2}}}}\]is equals to      [Roorkee 1999]
  A)
                  \[\frac{1}{2}\log 2\]        done clear
  B)
                  \[x=\frac{3\pi }{4}\] done clear
  C)
                  \[\pi /4\]              done clear
  D)
                  \[\pi /2\] done clear
  View Solution play_arrow

• question_answer8)
  \[\underset{n\to \infty }{\mathop{\lim }}\,\,\left[ \frac{1}{n}+\frac{1}{\sqrt{{{n}^{2}}+n}}+\frac{1}{\sqrt{{{n}^{2}}+2n}}+.....+\frac{1}{\sqrt{{{n}^{2}}+(n-1)n}} \right]\] is equal to            [RPET 2000]
  A)
                  \[2+2\sqrt{2}\]  done clear
  B)
                  \[2\sqrt{2}-2\] done clear
  C)
                  \[2\sqrt{2}\]       done clear
  D)
                  2 done clear
  View Solution play_arrow

• question_answer9)
  \[\underset{n\to \infty }{\mathop{\lim }}\,\frac{{{1}^{p}}+{{2}^{p}}+{{3}^{p}}+.....+{{n}^{p}}}{{{n}^{p+1}}}=\]   [AIEEE 2002]
  A)
                  \[\frac{1}{p+1}\]              done clear
  B)
                  \[\frac{1}{1-p}\] done clear
  C)
                  \[\frac{1}{p}-\frac{1}{p-1}\]        done clear
  D)
                  \[\underset{x\to 0-}{\mathop{\lim }}\,f(x)=0\] done clear
  View Solution play_arrow

• question_answer10)
  \[\underset{n\to \infty }{\mathop{\lim }}\,\sum\limits_{r=1}^{n}{\frac{1}{n}{{e}^{\frac{r}{n}}}}\]is         [AIEEE 2004]
  A)
                  \[e+1\] done clear
  B)
                  \[e-1\] done clear
  C)
                  \[1-e\]  done clear
  D)
                  \[e\] done clear
  View Solution play_arrow

• question_answer11)
  \[\int_{0}^{\infty }{\frac{\log \,(1+{{x}^{2}})}{1+{{x}^{2}}}}\,dx=\]
  A)
                  \[\pi \log \frac{1}{2}\]    done clear
  B)
                  \[\pi \log 2\] done clear
  C)
                  \[2\pi \log \frac{1}{2}\]  done clear
  D)
                  \[2\pi \log 2\] done clear
  View Solution play_arrow

• question_answer12)
  \[\int_{0}^{\pi /2}{{{\sin }^{2}}x{{\cos }^{3}}x}\,dx=\]                                          [RPET 1984, 2003]
  A)
                  0              done clear
  B)
                  \[\frac{2}{15}\] done clear
  C)
                  \[\frac{4}{15}\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer13)
  \[\int_{-\pi /2}^{\pi /2}{{{\sin }^{2}}x\,dx=}\]
  A)
                  \[\pi \]  done clear
  B)
                  \[\frac{\pi }{2}\] done clear
  C)
                  \[\frac{\pi }{2}-\frac{1}{2}\]         done clear
  D)
                  \[\pi -1\] done clear
  View Solution play_arrow

• question_answer14)
  The correct evaluation of \[\int_{0}^{\pi }{\left| \,{{\sin }^{4}}x\, \right|\,dx}\] is             [MP PET 1993]
  A)
                  \[\frac{8\pi }{3}\]             done clear
  B)
                  \[\frac{2\pi }{3}\] done clear
  C)
                  \[\frac{4\pi }{3}\]             done clear
  D)
                  \[\frac{3\pi }{8}\] done clear
  View Solution play_arrow

• question_answer15)
  \[\int_{0}^{\pi /2}{{{\cos }^{2}}x\,dx=}\]
  A)
                  \[1-\frac{\pi }{4}\]           done clear
  B)
                  \[1+\frac{\pi }{4}\] done clear
  C)
                  \[\frac{\pi }{4}\]               done clear
  D)
                  \[\frac{\pi }{2}\] done clear
  View Solution play_arrow

• question_answer16)
  The points of intersection of                 \[{{F}_{1}}(x)=\int_{2}^{x}{(2t-5)\,dt}\] and \[{{F}_{2}}(x)=\int_{0}^{x}{2t\,dt,}\] are      [IIT Screening]
  A)
                  \[\left( \frac{6}{5},\,\frac{36}{25} \right)\]            done clear
  B)
                  \[\left( \frac{2}{3},\,\frac{4}{9} \right)\] done clear
  C)
                  \[\left( \frac{1}{3},\,\frac{1}{9} \right)\] done clear
  D)
                  \[\left( \frac{1}{5},\,\frac{1}{25} \right)\] done clear
  View Solution play_arrow

• question_answer17)
  \[\int_{0}^{\infty }{\frac{{{x}^{2}}\,dx}{({{x}^{2}}+{{a}^{2}})({{x}^{2}}+{{b}^{2}})}}=\]
  A)
                  \[\frac{\pi }{2(a-b)}\]     done clear
  B)
                  \[\frac{\pi }{2(b-a)}\] done clear
  C)
                  \[\frac{\pi }{(a+b)}\]       done clear
  D)
                  \[\frac{\pi }{2(a+b)}\] done clear
  View Solution play_arrow

• question_answer18)
  \[\int_{0}^{\infty }{\frac{{{x}^{3}}\,dx}{{{({{x}^{2}}+4)}^{2}}}=}\]
  A)
                  0              done clear
  B)
                  \[\infty \] done clear
  C)
                  \[\frac{1}{2}\]    done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer19)
  \[\int_{0}^{\pi /2}{{{\sin }^{2m}}x\,dx=}\]
  A)
                  \[\frac{2\,\,m\,\,!}{{{({{2}^{m}}.\,m\,\,!)}^{2}}}.\frac{\pi }{2}\]  done clear
  B)
                  \[\frac{(2m)\,\,!}{{{({{2}^{m}}.\,m\,\,!)}^{2}}}.\frac{\pi }{2}\] done clear
  C)
                  \[\frac{2m\,\,!}{{{2}^{m}}.\,{{(m\,\,!)}^{2}}}.\frac{\pi }{2}\]         done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer20)
  \[\int_{0}^{\pi /2}{{{\sin }^{5}}x\,dx=}\]
  A)
                  \[\frac{8}{15}\] done clear
  B)
                  \[\frac{4}{15}\] done clear
  C)
                  \[\frac{8\sqrt{\pi }}{15}\]             done clear
  D)
                  \[\frac{8\pi }{15}\] done clear
  View Solution play_arrow

• question_answer21)
  \[\int_{0}^{\infty }{\frac{x\,dx}{(1+x)(1+{{x}^{2}})}}=\]
  A)
                  \[\frac{\pi }{4}\]               done clear
  B)
                  \[\frac{\pi }{3}\] done clear
  C)
                  \[\frac{\pi }{6}\]               done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer22)
  If \[\varphi (x)=\int_{1/x}^{\sqrt{x}}{\sin ({{t}^{2}})\,dt,}\] then \[{\varphi }'(1)=\]
  A)
                  \[\sin 1\]              done clear
  B)
                  \[2\sin 1\] done clear
  C)
                  \[\frac{3}{2}\sin 1\]         done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer23)
  The value of integral \[\int_{0}^{1}{\frac{{{x}^{b}}-1}{\log x}}\,dx\] is
  A)
                  \[\log b\]             done clear
  B)
                  \[2\log (b+1)\] done clear
  C)
                  \[3\log b\]           done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer24)
  The value of  the integral \[\int_{-1}^{1}{\frac{d}{dx}\left( {{\tan }^{-1}}\frac{1}{x} \right)}\,dx\] is
  A)
                  \[\frac{\pi }{2}\]               done clear
  B)
                  \[\frac{\pi }{4}\] done clear
  C)
                  \[-\frac{\pi }{2}\]              done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer25)
  The least value of the function \[F(x)=\] \[\int_{5\pi /4}^{x}{(3\sin u+4\cos u)\,du}\] on the interval \[\left[ \frac{5\pi }{4},\,\,\frac{4\pi }{3} \right]\] is
  A)
                  \[\sqrt{3}+\frac{3}{2}\] done clear
  B)
                  \[-2\sqrt{3}+\frac{3}{2}+\frac{1}{\sqrt{2}}\] done clear
  C)
                  \[\frac{3}{2}+\frac{1}{\sqrt{2}}\]               done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer26)
  \[\int_{0}^{\infty }{{{e}^{-2x}}(\sin 2x+\cos 2x)\,dx=}\]
  A)
                  1              done clear
  B)
                  0 done clear
  C)
                  \[\frac{1}{2}\]    done clear
  D)
                  \[\infty \] done clear
  View Solution play_arrow

• question_answer27)
  \[\int_{0}^{b-c}{\,\,{f}''(x+a)\,dx=}\]                                      [SCRA 1990]
  A)
                  \[{f}'(a)-{f}'(b)\]               done clear
  B)
                  \[{f}'(b-c+a)-{f}'(a)\] done clear
  C)
                  \[{f}'(b+c-a)+{f}'(a)\]      done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer28)
  The greatest value of the function \[F(x)=\int_{1}^{x}{\,\,|t|\,dt}\] on the interval \[\left[ -\frac{1}{2},\,\,\frac{1}{2} \right]\] is given by                                         [IIT Screening]
  A)
                  \[\frac{3}{8}\]    done clear
  B)
                  \[-\frac{1}{2}\] done clear
  C)
                  \[-\frac{3}{8}\]  done clear
  D)
                  \[\frac{2}{5}\] done clear
  View Solution play_arrow

• question_answer29)
  \[\int_{-\pi /2}^{\pi /2}{{{\sin }^{2}}x{{\cos }^{2}}x(\sin x+\cos x)\,dx=}\]             [EAMCET 1992]
  A)
                  \[\frac{2}{15}\] done clear
  B)
                  \[\frac{4}{15}\] done clear
  C)
                  \[\frac{6}{15}\] done clear
  D)
                  \[\frac{8}{15}\] done clear
  View Solution play_arrow

• question_answer30)
  \[\int_{0}^{\infty }{\frac{dx}{{{\left( x+\sqrt{{{x}^{2}}+1} \right)}^{3}}}}=\]                                          [EAMCET 1992]
  A)
                  \[\frac{3}{8}\]    done clear
  B)
                  \[\frac{1}{8}\] done clear
  C)
                  \[-\frac{3}{8}\]  done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer31)
  The derivative of \[F(x)=\int_{{{x}^{2}}}^{{{x}^{3}}}{\frac{1}{\log t}\,dt}\], \[(x>0)\] is
  A)
                  \[\frac{1}{3\log x}-\frac{1}{2\log x}\]      done clear
  B)
                  \[\frac{1}{3\log x}\] done clear
  C)
                  \[\frac{3{{x}^{2}}}{3\log x}\]       done clear
  D)
                  \[{{(\log x)}^{-1}}.x(x-1)\] done clear
  View Solution play_arrow

• question_answer32)
  If \[f(x)=\int_{{{x}^{2}}}^{{{x}^{2}}+1}{{{e}^{-{{t}^{2}}}}}dt,\] then \[f(x)\] increases in [IIT Screening 2003]
  A)
                  \[(2,\,\,2)\]         done clear
  B)
                  No value of \[x\] done clear
  C)
                  \[(0,\,\,\infty )\]               done clear
  D)
                  \[(-\infty ,\,\,0)\] done clear
  View Solution play_arrow

• question_answer33)
  If \[f(x)=\int_{{{x}^{2}}}^{{{x}^{4}}}{\sin \sqrt{t}\,dt,}\] then \[{f}'(x)\] equals
  A)
                  \[\sin {{x}^{2}}-\sin x\]  done clear
  B)
                  \[4{{x}^{3}}\sin {{x}^{2}}-2x\sin x\] done clear
  C)
                  \[{{x}^{4}}\sin {{x}^{2}}-x\sin x\]              done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer34)
  If \[F(x)=\frac{1}{{{x}^{2}}}\int_{4}^{x}{(4{{t}^{2}}-2{F}'(t))\,dt,}\] then \[{F}'(4)\] equals
  A)
                  32           done clear
  B)
                  \[\frac{32}{3}\] done clear
  C)
                  \[\frac{32}{9}\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer35)
  The value of the integral \[\sum\limits_{k=1}^{n}{\int_{0}^{1}{f(k-1+x)\,dx}}\] is
  A)
                  \[\int_{0}^{1}{f(x)\,dx}\]              done clear
  B)
                  \[\int_{0}^{2}{f(x)\,dx}\] done clear
  C)
                  \[\int_{0}^{n}{f(x)\,dx}\]             done clear
  D)
                  \[n\int_{0}^{1}{f(x)\,dx}\] done clear
  View Solution play_arrow

• question_answer36)
  The value of \[\int_{a}^{a+(\pi /2)}{({{\sin }^{4}}x+{{\cos }^{4}}x)\,dx}\] is
  A)
                  Independent of \[a\]     done clear
  B)
                  \[a\,{{\left( \frac{\pi }{2} \right)}^{2}}\] done clear
  C)
                  \[\frac{3\pi }{8}\]             done clear
  D)
                  \[\frac{3\pi {{a}^{2}}}{8}\] done clear
  View Solution play_arrow

• question_answer37)
  \[\int_{0}^{\pi }{{{\sin }^{5}}\left( \frac{x}{2} \right)\,dx}\] equals                            [Kurukshetra CEE 1996]
  A)
                  \[\frac{16}{15}\]               done clear
  B)
                  \[\frac{32}{15}\] done clear
  C)
                  \[\frac{8}{15}\] done clear
  D)
                  \[\frac{5}{6}\] done clear
  View Solution play_arrow

• question_answer38)
  If \[\int_{{}}^{{}}{f(x)\,dx}=x{{e}^{-\log |x|}}+f(x),\] then \[f(x)\] is        [MP PET 1997]
  A)
                  1              done clear
  B)
                  0 done clear
  C)
                  \[c{{e}^{x}}\]     done clear
  D)
                  \[\log x\] done clear
  View Solution play_arrow

• question_answer39)
  \[\int_{0}^{\pi /2}{{{\sin }^{4}}x{{\cos }^{6}}x\,dx}\] equals                                         [RPET 1999]
  A)
                  \[\frac{5\pi }{512}\]        done clear
  B)
                  \[\frac{3\pi }{512}\] done clear
  C)
                  \[\frac{\pi }{512}\]           done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer40)
  The value of \[\int_{\,0}^{\,\pi /2}{{{\left( \sqrt{\sin \theta }\cos \theta  \right)}^{3}}d\theta }\] is           [AMU 1999]
  A)
                  2/9         done clear
  B)
                  2/15 done clear
  C)
                  8/45       done clear
  D)
                  5/2 done clear
  View Solution play_arrow

• question_answer41)
  \[\int_{\,0}^{\,\infty }{\,\log \left( x+\frac{1}{x} \right)\frac{dx}{1+{{x}^{2}}}}\] is equal to             [RPET 2000, 02]
  A)
                  \[\pi \log 2\]       done clear
  B)
                  \[-\pi \log 2\] done clear
  C)
                  \[(\pi /2)\log 2\]               done clear
  D)
                  \[-(\pi /2)\log 2\] done clear
  View Solution play_arrow

• question_answer42)
  \[\int_{\,0}^{\,\infty }{\frac{x\ln x\,dx}{{{(1+{{x}^{2}})}^{2}}}}\] is equal to                                          [AMU 2000]
  A)
                  0              done clear
  B)
                  1 done clear
  C)
                  \[\infty \]            done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer43)
  If \[f(t)=\int_{\,-t}^{\,t}{\frac{dx}{1+{{x}^{2}}},}\] then \[{f}'(1)\] is                         [Roorkee 2000]
  A)
                  Zero       done clear
  B)
                  2/3 done clear
  C)
                  \[-\,1\] done clear
  D)
                  1 done clear
  View Solution play_arrow

• question_answer44)
  If \[F(x)=\int_{{{x}^{2}}}^{{{x}^{3}}}{\log t\,dt,\,\,(x>0),}\] then \[{F}'(x)=\]        [MP PET 2001]
  A)
                  \[(9{{x}^{2}}-4x)\log x\]                done clear
  B)
                  \[(4x-9{{x}^{2}})\log x\] done clear
  C)
                  \[(9{{x}^{2}}+4x)\log x\]               done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer45)
  \[\int_{\,-\pi /2}^{\,\pi /2}{{{\sin }^{4}}x{{\cos }^{6}}x\,dx=}\]                                    [EAMCET 2002]
  A)
                  \[\frac{3\pi }{64}\]           done clear
  B)
                  \[\frac{3\pi }{572}\] done clear
  C)
                  \[\frac{3\pi }{256}\]        done clear
  D)
                  \[\frac{3\pi }{128}\] done clear
  View Solution play_arrow

• question_answer46)
  \[\int_{\,0}^{\,1}{\frac{d}{dx}\left[ {{\sin }^{-1}}\left( \frac{2x}{1+{{x}^{2}}} \right) \right]\,dx}\] is equal to  [Kerala (Engg.) 2002]
  A)
                  0              done clear
  B)
                  \[\pi \] done clear
  C)
                  \[\pi /2\]              done clear
  D)
                  \[\pi /4\] done clear
  View Solution play_arrow

• question_answer47)
  Let \[f(x)=\int_{\,1}^{\,x}{\sqrt{2-{{t}^{2}}}dt}\]. Then real roots of the equation \[{{x}^{2}}-{f}'(x)=0\] are                                         [IIT Screening 2002]
  A)
                  \[\pm 1\]             done clear
  B)
                  \[\pm \frac{1}{\sqrt{2}}\] done clear
  C)
                  \[\pm \,\,\frac{1}{2}\]   done clear
  D)
                  0 and 1 done clear
  View Solution play_arrow

• question_answer48)
  \[\int_{\,0}^{\,\infty }{\frac{xdx}{(1+x)(1+{{x}^{2}})}=}\]                                              [Karnataka CET 2003]
  A)
                  0              done clear
  B)
                  \[\pi /2\] done clear
  C)
                  \[\pi /4\]              done clear
  D)
                  1 done clear
  View Solution play_arrow

• question_answer49)
  \[\int_{0}^{a}{{{x}^{4}}\sqrt{{{a}^{2}}-{{x}^{2}}}}\,dx=\]
  A)
                  \[\frac{\pi }{32}\]             done clear
  B)
                  \[\frac{\pi }{32}{{a}^{6}}\] done clear
  C)
                  \[\frac{\pi }{16}{{a}^{6}}\]            done clear
  D)
                  \[\frac{\pi }{8}{{a}^{6}}\] done clear
  View Solution play_arrow

• question_answer50)
  \[\int_{0}^{a}{x{{(2ax-{{x}^{2}})}^{\frac{3}{2}}}\,dx=}\]
  A)
                  \[{{a}^{5}}\left[ \frac{3\pi }{16}-1 \right]\]            done clear
  B)
                  \[{{a}^{5}}\left[ \frac{3\pi }{16}+1 \right]\] done clear
  C)
                  \[{{a}^{5}}\left[ \frac{3\pi }{16}-\frac{1}{5} \right]\]        done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer51)
  \[\int_{0}^{a}{{{x}^{2}}{{({{a}^{2}}-{{x}^{2}})}^{3/2}}dx=}\]
  A)
                  \[\frac{\pi {{a}^{6}}}{32}\]            done clear
  B)
                  \[\frac{2{{a}^{5}}}{15}\] done clear
  C)
                  \[\frac{{{a}^{6}}}{32}\]  done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer52)
  Let\[\frac{d}{dx}F(x)=\left( \frac{{{e}^{\sin x}}}{x} \right)\,;\,x>0\]. If \[\int_{\,1}^{\,4}{\frac{3}{x}{{e}^{\sin {{x}^{3}}}}dx=F(k)-F(1)}\], then one of the possible value of k, is               [AIEEE 2003]
  A)
                  15           done clear
  B)
                  16 done clear
  C)
                  63           done clear
  D)
                  64 done clear
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• question_answer53)
  If \[f(x)=\int_{0}^{x}{t\sin t\,dt\,,}\] then \[{f}'(x)=\]                                 [MNR 1982; Karnataka CET 1999]
  A)
                  \[\cos x+x\sin x\]             done clear
  B)
                  \[x\sin x\] done clear
  C)
                  \[x\cos x\]          done clear
  D)
                  None of these done clear
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• question_answer54)
  \[\underset{n\to \infty }{\mathop{\text{lim}\,}}\,\left[ \frac{1}{{{n}^{2}}}{{\sec }^{2}}\frac{1}{{{n}^{2}}}+\frac{2}{{{n}^{2}}}{{\sec }^{2}}\frac{4}{{{n}^{2}}}+.....+\frac{1}{n}{{\sec }^{2}}1 \right]\] equals [AIEEE 2005]
  A)
                  \[\tan 1\]             done clear
  B)
                  \[\frac{1}{2}\tan 1\]        done clear
  C)
                  \[\frac{1}{2}\sec 1\]        done clear
  D)
                  \[\frac{1}{2}\text{cosec}1\] done clear
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