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question_answer1)
\[\int \frac{dx}{x({{x}^{n}}+1)}\] is equal to
A)
\[\frac{1}{n}\log \left( \frac{{{x}^{n}}}{{{x}^{n}}+1} \right)+c\] done
clear
B)
\[\frac{1}{n}\log \left( \frac{{{x}^{n}}+1}{{{x}^{n}}} \right)+c\] done
clear
C)
\[\log \left( \frac{{{x}^{n}}}{{{x}^{n}}+1} \right)+c\] done
clear
D)
none of these done
clear
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question_answer2)
If \[{{I}_{n}}=\int {{(ln\,x)}^{n}}dx\], then \[{{I}_{n}}+n{{I}_{n-1}}\]=
A)
\[\frac{{{(ln\,x)}^{n}}}{x}+C\] done
clear
B)
\[x{{(ln\,x)}^{n-1}}+C\] done
clear
C)
\[x{{(ln\,x)}^{n}}+C\] done
clear
D)
none of these done
clear
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question_answer3)
\[\int \frac{{{\sin }^{8}}x-{{\cos }^{8}}x}{1-2{{\sin }^{2}}x{{\cos }^{2}}x}dx\] is equal to
A)
\[\frac{1}{2}\sin 2x+C\] done
clear
B)
\[-\frac{1}{2}\sin 2x+C\] done
clear
C)
\[-\frac{1}{2}\sin x+C\] done
clear
D)
\[-{{\sin }^{2}}x+C\] done
clear
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question_answer4)
\[\int \frac{2\sin x}{(3+sin2x)}dx\]is equal to
A)
\[\frac{1}{2}\ln \left| \frac{2+\sin x-\cos x}{2-\sin x+\cos x} \right|-\frac{1}{\sqrt{2}}{{\tan }^{-1}}\left( \frac{\sin x+\cos x}{\sqrt{2}} \right)+c\] done
clear
B)
\[\frac{1}{2}\ln \left| \frac{2+\sin x-\cos x}{2-\sin x+\cos x} \right|-\frac{1}{2\sqrt{2}}{{\tan }^{-1}}\left( \frac{\sin x+\cos x}{\sqrt{2}} \right)+c\] done
clear
C)
\[\frac{1}{4}\ln \left| \frac{2+\sin x-\cos x}{2-\sin x+\cos x} \right|-\frac{1}{\sqrt{2}}{{\tan }^{-1}}\left( \frac{\sin x+\cos x}{\sqrt{2}} \right)+c\] done
clear
D)
none of these done
clear
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question_answer5)
If \[\int {{x}^{5}}{{(1+{{x}^{3}})}^{2/3}}dx=A{{(1+{{x}^{3}})}^{8/3}}+B{{(1+{{x}^{3}})}^{5/3}}+c\],then
A)
\[A=\frac{1}{4},B=\frac{1}{5}\] done
clear
B)
\[A=\frac{1}{8},B=-\frac{1}{5}\] done
clear
C)
\[A=-\frac{1}{8},B=\frac{1}{5}\] done
clear
D)
none of these done
clear
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question_answer6)
If\[\int \frac{3\sin x+2\cos x}{3\cos x+2\sin x}dx=ax+b\] ln\[\left| 2\sin x+3\cos x \right|+C\], then
A)
\[a=-\frac{12}{13},b=\frac{15}{39}\] done
clear
B)
\[a=-\frac{7}{13},b=\frac{6}{13}\] done
clear
C)
\[a=\frac{12}{13},b=-\frac{15}{39}\] done
clear
D)
\[a=-\frac{7}{13},b=-\frac{6}{13}\] done
clear
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question_answer7)
\[\int \frac{\cos 4x-1}{\cot x-\tan x}dx\]is equal to
A)
\[\frac{1}{2}\ln \left| \sec 2x \right|-\frac{1}{4}{{\cos }^{2}}2x+c\] done
clear
B)
\[\frac{1}{2}\ln \left| \sec 2x \right|+\frac{1}{4}{{\cos }^{2}}x+c\] done
clear
C)
\[\frac{1}{2}\ln \left| \cos 2x \right|-\frac{1}{4}{{\cos }^{2}}2x+c\] done
clear
D)
\[\frac{1}{2}\ln \left| \cos 2x \right|+\frac{1}{4}{{\cos }^{2}}x+c\] done
clear
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question_answer8)
\[\int \frac{\sqrt{x-1}}{x\sqrt{x+1}}dx\]is equal to
A)
\[\ln \,\left| x-\sqrt{{{x}^{2}}-1} \right|-{{\tan }^{-1}}x+c\] done
clear
B)
\[\ln \,\left| x+\sqrt{{{x}^{2}}-1} \right|-{{\tan }^{-1}}x+c\] done
clear
C)
\[\ln \,\left| x-\sqrt{{{x}^{2}}-1} \right|-{{\sec }^{-1}}x+c\] done
clear
D)
\[\ln \,\left| x+\sqrt{{{x}^{2}}-1} \right|-{{\sec }^{-1}}x+c\] done
clear
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question_answer9)
If \[\int \frac{\sin x}{\sin (x-\alpha )}dx=Ax+B\,\,\log \sin (x-\alpha )+c,\] then the value of (A, B) is
A)
\[(sin\alpha ,\,cos\alpha )\] done
clear
B)
\[(cos\alpha ,\,\sin \alpha )\] done
clear
C)
\[(-\sin \alpha ,\,\cos \alpha )\] done
clear
D)
\[(-\cos \alpha ,\,\sin \alpha )\] done
clear
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question_answer10)
\[\int \frac{dx}{\cos x+\sqrt{3}\sin x}\] is equal to
A)
\[\frac{1}{2}\log \,\tan \left( \frac{x}{2}+\frac{\pi }{12} \right)+c\] done
clear
B)
\[\frac{1}{2}\log \,\,\tan \left( \frac{x}{2}-\frac{\pi }{12} \right)+c\] done
clear
C)
\[\log \,\,\tan \left( \frac{x}{2}+\frac{\pi }{12} \right)+c\] done
clear
D)
\[\log \,\tan \left( \frac{x}{2}-\frac{\pi }{12} \right)+c\] done
clear
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question_answer11)
\[\int {{\left\{ \frac{\log x-1}{1+{{(log\,x)}^{2}}} \right\}}^{2}}\]dx is equal to
A)
\[\frac{\log \,x}{{{(log\,x)}^{2}}+1}+c\] done
clear
B)
\[\frac{x}{{{x}^{2}}+1}+c\] done
clear
C)
\[\frac{x{{e}^{x}}}{1+{{x}^{2}}}+c\] done
clear
D)
\[\frac{x}{{{(log\,x)}^{2}}+1}+c\] done
clear
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question_answer12)
Let \[f(0)=0\] and \[\int\limits_{0}^{2}{f'(2t){{e}^{f(2t)}}dt=5}\]. Then the value of f(4) is
A)
log 2 done
clear
B)
log 7 done
clear
C)
log 11 done
clear
D)
log 13 done
clear
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question_answer13)
Which of the following is incorrect?
A)
\[\int_{a+c}^{b+c}{f(x)dx=\int_{a}^{b}{f(x+c)dx}}\] done
clear
B)
\[\int_{ac}^{bc}{f(x)dx=c\int_{a}^{b}{f(cx)dx}}\] done
clear
C)
\[\int_{-a}^{a}{f(x)dx=\frac{1}{2}\int_{-a}^{a}{f(x)+f(-x)dx}}\] done
clear
D)
None of these done
clear
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question_answer14)
The value of \[\int\limits_{1}^{e}{\frac{1+{{x}^{2}}\ln \,x}{x+{{x}^{2}}\ln \,x}dx}\]is
A)
\[e\] done
clear
B)
\[\ln \,(1+e)\] done
clear
C)
\[e+\ln (1+e)\] done
clear
D)
\[e-\ln (1+e)\] done
clear
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question_answer15)
Given \[\int\limits_{0}^{\pi /2}{\frac{dx}{1+\sin x+\cos x}=\log \,2.}\] Then the value of the definite integral \[\int\limits_{0}^{\pi /2}{\frac{\sin x}{1+\sin x+\cos x}dx}\]is equal to
A)
\[\frac{1}{2}\log 2\] done
clear
B)
\[\frac{\pi }{2}-\log 2\] done
clear
C)
\[\frac{\pi }{4}-\frac{1}{2}\log 2\] done
clear
D)
\[\frac{\pi }{2}+\log 2\] done
clear
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question_answer16)
If \[\int_{0}^{1}{{{\cot }^{-1}}(1-x+{{x}^{2}})dx=\lambda \int_{0}^{1}{{{\tan }^{-1}}xdx,}}\] then \[\lambda \] is equal to
A)
1 done
clear
B)
2 done
clear
C)
3 done
clear
D)
4 done
clear
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question_answer17)
If \[f(x)=\frac{{{e}^{x}}}{1+{{e}^{x}}},\,\,\,{{I}_{1}}=\int\limits_{f(-a)}^{f(a)}{xg\,(x(1-x))dx,}\] and \[{{I}_{2}}=\int\limits_{f(-a)}^{f(a)}{g(x(1-x))dx,}\] then the value of \[\frac{{{I}_{2}}}{{{I}_{1}}}\]is
A)
\[-\,1\] done
clear
B)
\[-\,2\] done
clear
C)
2 done
clear
D)
1 done
clear
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question_answer18)
If \[\int_{1}^{2}{{{e}^{{{x}^{2}}}}dx=a}\], then \[\int_{e}^{{{e}^{4}}}{\sqrt{\ln x}}\,dx\] is equal to
A)
\[2{{e}^{4}}-2e-a\] done
clear
B)
\[2{{e}^{4}}-e-a\] done
clear
C)
\[2{{e}^{4}}-e-2a\] done
clear
D)
\[{{e}^{4}}-e-a\] done
clear
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question_answer19)
\[\int\limits_{0}^{\infty }{\left( \frac{\pi }{1+{{\pi }^{2}}{{x}^{2}}}-\frac{1}{1+{{x}^{2}}} \right)\log x\,dx}\]is equal to
A)
\[-\frac{\pi }{2}\ln \pi \] done
clear
B)
0 done
clear
C)
\[\frac{\pi }{2}\ln 2\] done
clear
D)
none of these done
clear
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question_answer20)
If \[A=\int_{0}^{\pi }{\frac{\operatorname{cosx}}{{{(x+2)}^{2}}}dx,}\] then \[A=\int_{0}^{\pi /2}{\frac{\sin 2x}{x+1}dx,}\] is equal to
A)
\[\frac{1}{2}+\frac{1}{\pi +2}-A\] done
clear
B)
\[\frac{1}{\pi +2}-A\] done
clear
C)
\[1+\frac{1}{\pi +2}-A\] done
clear
D)
\[A-\frac{1}{2}-\frac{1}{\pi +2}\] done
clear
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question_answer21)
If \[y=\int{\frac{dx}{{{(1+{{x}^{2}})}^{\frac{3}{2}}}}}\]and y=0 when x=0, the value of y when x=1 is _____.
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question_answer22)
If \[\int{\frac{2\tan x}{\tan x-2}dx=x+a}\] \[\ln \left( \text{sin}\,x-2\,\text{cos}\,x \right)+x\]then a = ______.
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question_answer23)
If P(x) is a polynomial of the least degree that has a maximum equal to 6 at \[x=1,\] and a minimum equal to 2 at \[x=3,\] then \[\int_{0}^{1}{P(x)dx}\] equal _______.
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question_answer24)
\[{{I}_{n}}=\int\limits_{0}^{\pi /4}{{{\tan }^{n}}x\,dx,}\] then \[\underset{n\to \infty }{\mathop{\lim }}\,n\,[{{I}_{n}}+{{I}_{n+2}}]\]dx is equal to _______.
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question_answer25)
\[\int\limits_{-\pi }^{\pi }{\frac{2x(1+sinx)}{1+{{\cos }^{2}}x}dx}\] is equal to ______.
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