-
question_answer1)
\[\int{\frac{\sqrt{x}}{1+\sqrt[4]{{{x}^{3}}}}}dx\] is equal to
A)
\[\frac{4}{3}\left[ 1+{{x}^{3/4}}+\log (1+{{x}^{3/4}}) \right]+C\] done
clear
B)
\[\frac{4}{3}\left[ 1+{{x}^{3/4}}-\log (1+{{x}^{3/4}}) \right]+C\] done
clear
C)
\[\frac{4}{3}\left[ 1-{{x}^{3/4}}+\log (1+{{x}^{3/4}}) \right]+C\] done
clear
D)
None of these done
clear
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question_answer2)
\[\int{\frac{({{x}^{2}}-1)}{x\sqrt{{{x}^{4}}+3{{x}^{2}}+1}}dx}\] is equal to
A)
\[\log \left| x+\frac{1}{x}+\sqrt{{{x}^{2}}+\frac{1}{{{x}^{2}}}+3} \right|+C\] done
clear
B)
\[\log \left| x-\frac{1}{x}+\sqrt{{{x}^{2}}+\frac{1}{{{x}^{2}}}-3} \right|+C\] done
clear
C)
\[\log \left| x+\sqrt{{{x}^{2}}+3} \right|+C\] done
clear
D)
None of these done
clear
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question_answer3)
\[\int{\frac{(1+x){{e}^{x}}}{\cot (x{{e}^{x}})}dx}\] is equal to
A)
\[\log \left| \cos (x{{e}^{x}}) \right|+C\] done
clear
B)
\[\log \left| \cot (x{{e}^{x}}) \right|+C\] done
clear
C)
\[\log \left| sec(x{{e}^{-x}}) \right|+C\] done
clear
D)
\[\log \left| sec(x{{e}^{x}}) \right|+C\] done
clear
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question_answer4)
\[\int{\frac{dx}{\cos x+\sqrt{3}\sin x}}\] equals
A)
\[\log \tan \left( \frac{x}{2}+\frac{\pi }{12} \right)+C\] done
clear
B)
\[\log \tan \left( \frac{x}{2}-\frac{\pi }{12} \right)+C\] done
clear
C)
\[\frac{1}{2}\log \tan \left( \frac{x}{2}+\frac{\pi }{12} \right)+C\] done
clear
D)
\[\frac{1}{2}\log \tan \left( \frac{x}{2}-\frac{\pi }{12} \right)+C\] done
clear
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question_answer5)
If \[I=\int{\frac{1}{2p}\sqrt{\frac{p-1}{p+1}}dp=f(p)+c}\], then f(p) is equal to:
A)
\[\frac{1}{2}\ell n\left[ p-\sqrt{{{p}^{2}}-1} \right]\] done
clear
B)
\[\frac{1}{2}{{\cos }^{-1}}p+\frac{1}{2}{{\sec }^{-1}}p\] done
clear
C)
\[\ell n\sqrt{p+\sqrt{{{p}^{2}}-1}}-\frac{1}{2}{{\sec }^{-1}}p\] done
clear
D)
None of the above. done
clear
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question_answer6)
Let\[f(x)=\int{{{e}^{x}}(x-1)(x-2)}dx\]. Then f decreases in the interval
A)
\[(-\infty ,-2)\] done
clear
B)
\[(-2,-1)\] done
clear
C)
\[(1,\,\,2)\] done
clear
D)
\[(2,+\infty )\] done
clear
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question_answer7)
Let \[f:R\to R\] is differentiable function and \[f(1)=4,\] then the value of \[\underset{x\to 1}{\mathop{\lim }}\,\int\limits_{0}^{f(x)}{\frac{2tdt}{x-1}}\] is
A)
\[8f'(1)\] done
clear
B)
\[4f'(1)\] done
clear
C)
\[2f'(1)\] done
clear
D)
\[f'(1)\] done
clear
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question_answer8)
\[\int{{{\left( x+\frac{1}{x} \right)}^{n+5}}\left( \frac{{{x}^{2}}-1}{{{x}^{2}}} \right)dx}\] is equal to:
A)
\[\frac{{{\left( x+\frac{1}{x} \right)}^{n+6}}}{n+6}+c\] done
clear
B)
\[{{\left[ \frac{{{x}^{2}}+1}{{{x}^{2}}} \right]}^{n+6}}(n+6)+c\] done
clear
C)
\[{{\left[ \frac{x}{{{x}^{2}}+1} \right]}^{n+6}}(n+6)+c\] done
clear
D)
None of these done
clear
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question_answer9)
Find the value of \[\int\limits_{0}^{9}{[\sqrt{x}+2]dx}\] where \[[\cdot ]\] is the greatest integer function:
A)
31 done
clear
B)
22 done
clear
C)
23 done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer10)
If \[\int{f(x)\sin x\cos x\,\,dx=\frac{1}{2({{b}^{2}}-{{a}^{2}})}{{\log }_{e}}(f(x))+A,}\]\[b\ne \pm a,\] then \[{{\{f(x)\}}^{-1}}\] is equal to
A)
\[{{a}^{2}}{{\sin }^{2}}x+{{b}^{2}}{{\cos }^{2}}x+C\] done
clear
B)
\[{{a}^{2}}{{\sin }^{2}}x-{{b}^{2}}{{\cos }^{2}}x+C\] done
clear
C)
\[{{a}^{2}}{{\cos }^{2}}x+{{b}^{2}}si{{n}^{2}}x+C\] done
clear
D)
\[{{a}^{2}}{{\cos }^{2}}x-{{b}^{2}}si{{n}^{2}}x+C\] done
clear
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question_answer11)
\[\int{\frac{{{x}^{2}}}{({{x}^{2}}+1)({{x}^{2}}+4)}dx}\] is equal to
A)
\[{{\tan }^{-1}}x+2{{\tan }^{-1}}\left( \frac{x}{2} \right)+C\] done
clear
B)
\[{{\tan }^{-1}}\left( \frac{x}{2} \right)-4{{\tan }^{-1}}x+C\] done
clear
C)
\[-\frac{1}{3}{{\tan }^{-1}}x+\frac{2}{3}{{\tan }^{-1}}\left( \frac{x}{2} \right)+C\] done
clear
D)
\[4{{\tan }^{-1}}\left( \frac{x}{2} \right)-2{{\tan }^{-1}}x+C\] done
clear
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question_answer12)
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)
\[-3\] done
clear
C)
\[-1\] done
clear
D)
\[2\] done
clear
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question_answer13)
\[\int{\sqrt{\frac{x}{1-x}}dx}\] is equal to
A)
\[{{\sin }^{-1}}\sqrt{x}+c\] done
clear
B)
\[{{\sin }^{-1}}\{\sqrt{x}-\sqrt{x(1-x)\}}+c\] done
clear
C)
\[{{\sin }^{-1}}\sqrt{x(1-x)}+c\] done
clear
D)
\[{{\sin }^{-1}}\sqrt{x}-\sqrt{x(1-x)}+c\] done
clear
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question_answer14)
What is \[\int{\frac{{{e}^{x}}(1+x)}{{{\cos }^{2}}\left( x{{e}^{x}} \right)}dx}\] equal to?
A)
\[x{{e}^{x}}+c\] done
clear
B)
\[\cos (x{{e}^{x}})+c\] done
clear
C)
\[\tan (x{{e}^{x}})+c\] done
clear
D)
\[x\cos ec(x{{e}^{x}})+c\] Where c is a constant of integration. done
clear
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question_answer15)
If\[f(x)=ln(x-\sqrt{1+{{x}^{2}}})\], then what is \[\int{f''(x)dx}\] equal to?
A)
\[\frac{1}{(x-\sqrt{1+{{x}^{2}}})}+c\] done
clear
B)
\[-\frac{1}{\sqrt{1+{{x}^{2}}}}+c\] done
clear
C)
\[-\sqrt{1+{{x}^{2}}}+c\] done
clear
D)
\[\text{ln}\,(x-\sqrt{1+{{x}^{2}}})+c\] done
clear
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question_answer16)
What is the value of\[\int\limits_{0}^{1}{(x-1){{e}^{-x}}dx}\]?
A)
0 done
clear
B)
e done
clear
C)
\[\frac{1}{e}\] done
clear
D)
\[\frac{-1}{e}\] done
clear
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question_answer17)
The value of \[\int_{0}^{{{\sin }^{2}}x}{{{\sin }^{-1}}\sqrt{t}\,\,dt}+\int_{0}^{{{\cos }^{2}}x}{{{\cos }^{-1}}\sqrt{t}dt}\] is
A)
\[\pi \] done
clear
B)
\[\frac{\pi }{2}\] done
clear
C)
\[\frac{\pi }{4}\] done
clear
D)
1 done
clear
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question_answer18)
\[\underset{n\to \infty }{\mathop{Lim}}\,{{\left\{ \frac{n!}{{{(kn)}^{n}}} \right\}}^{\frac{1}{n}}},\] where \[k\ne 0\] is a constant and \[n\in N\] is equal to
A)
ke done
clear
B)
\[{{k}^{-1}}e\] done
clear
C)
\[k{{e}^{-1}}\] done
clear
D)
\[{{k}^{-1}}{{e}^{-1}}\] done
clear
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question_answer19)
If \[{{I}_{1}}=\int\limits_{0}^{\pi }{xf({{\sin }^{3}}x+{{\cos }^{2}}x)dx}\] and \[{{I}_{2}}=\pi \int\limits_{0}^{\pi /2}{f({{\sin }^{3}}x+{{\cos }^{2}}x)dx}\], then
A)
\[{{I}_{1}}=2{{I}_{2}}\] done
clear
B)
\[2{{I}_{1}}={{I}_{2}}\] done
clear
C)
\[{{I}_{1}}={{I}_{2}}\] done
clear
D)
\[{{I}_{1}}+{{I}_{2}}=0\] done
clear
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question_answer20)
If \[{{l}^{r}}(x)\] means log log log ??.x, the log being repeated r times. then \[\int{\{xl(x){{l}^{2}}(x){{l}^{3}}(x)....{{l}^{r}}(x)\}{{-}^{1}}dx}\] is equal to
A)
\[{{l}^{r+1}}(x)+C\] done
clear
B)
\[\frac{{{l}^{r+1}}(x)}{r+1}+C\] done
clear
C)
\[{{l}^{r}}(x)+C\] done
clear
D)
None done
clear
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question_answer21)
If\[\int{{{\log }_{e}}\left( \sqrt{1-x}+\sqrt{1+x} \right)dx}\]\[=x{{\log }_{e}}\left( \sqrt{1-x}+\sqrt{1+x} \right)+g(x)+C\]. Then \[g(x)=\]
A)
\[x-{{\sin }^{-1}}x\] done
clear
B)
\[{{\sin }^{-1}}x-x\] done
clear
C)
\[x+{{\sin }^{-1}}x\] done
clear
D)
\[{{\sin }^{-1}}x-{{x}^{2}}\] done
clear
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question_answer22)
\[{{A}_{n}}=\int\limits_{0}^{\pi /2}{\frac{\sin (2n-1)x}{\sin x}}dx;{{B}_{n}}=\int\limits_{0}^{\pi /2}{{{\left( \frac{\sin nx}{\sin x} \right)}^{2}}dx;}\] For \[n\in N,\] then
A)
\[{{A}_{n+1}}={{A}_{n}},{{B}_{n+1}}-{{B}_{n}}={{A}_{n+1}}\] done
clear
B)
\[{{B}_{n+1}}={{B}_{n}}\] done
clear
C)
\[{{A}_{n+1}}-{{A}_{n}}={{B}_{n+1}}\] done
clear
D)
None of these done
clear
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question_answer23)
If \[\int{x\log \left( 1+\frac{1}{x} \right)dx}\]\[=f(x)\log (x+1)+g(x){{x}^{2}}+Lx+C\], then
A)
\[f(x)=\frac{1}{2}{{x}^{2}}\] done
clear
B)
\[g(x)=\log x\] done
clear
C)
\[L=1\] done
clear
D)
None of these done
clear
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question_answer24)
If \[\int{g(x)dx=g(x),}\] then\[\int{g(x)\{f(x)+f'(x)\}dx}\] is equal to
A)
\[g(x)f(x)-g(x)f'(x)+C\] done
clear
B)
\[g(x)f'(x)+C\] done
clear
C)
\[g(x)f(x)+C\] done
clear
D)
\[g(x){{f}^{2}}(x)+C\] done
clear
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question_answer25)
\[\int{\frac{dx}{\sin x(3+{{\cos }^{2}}x)}}\] is equal to
A)
\[\log \left| {{y}^{2}}-1 \right|-{{\tan }^{-1}}y+C\] done
clear
B)
\[{{\tan }^{-1}}\frac{y}{\sqrt{3}}+C\] done
clear
C)
\[\log \left| \frac{y-1}{y+1} \right|+C\] done
clear
D)
\[\frac{1}{4}\log \left| \frac{y-1}{y+1} \right|-\frac{1}{4\sqrt{3}}{{\tan }^{-1}}\frac{y}{\sqrt{3}}+C\] done
clear
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question_answer26)
\[\int\limits_{0}^{\pi }{xf(\sin \,\,x)dx}\] is equal to
A)
\[\pi \int\limits_{0}^{\pi }{f(cos\,\,x)dx}\] done
clear
B)
\[\pi \int\limits_{0}^{\pi }{f(sin\,\,x)dx}\] done
clear
C)
\[\frac{\pi }{2}\int\limits_{0}^{\pi /2}{f(sin\,\,x)dx}\] done
clear
D)
\[\pi \int\limits_{0}^{\pi /2}{f(cos\,\,x)dx}\] done
clear
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question_answer27)
\[\int{32{{x}^{3}}{{(\log \,\,x)}^{2}}dx}\] is equal to:
A)
\[8{{x}^{4}}{{(\log \,\,x)}^{2}}+C\] done
clear
B)
\[{{x}^{4}}\{8{{(\log \,\,x)}^{2}}-4(\log \,\,x)+1\}+C\] done
clear
C)
\[{{x}^{4}}\{8{{(\log \,\,x)}^{2}}-4(\log \,\,x)\}+C\] done
clear
D)
\[{{x}^{3}}\{{{(\log \,\,x)}^{2}}-2\log \,\,x\}+C\] done
clear
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question_answer28)
\[\left[ \sum\limits_{n=1}^{10}{\int\limits_{-2n-1}^{-2n}{{{\sin }^{27}}xdx}} \right]+\left[ \sum\limits_{n=1}^{10}{\int\limits_{2n}^{2n+1}{{{\sin }^{27}}}xdx} \right]=\]
A)
\[{{27}^{2}}\] done
clear
B)
\[-54\] done
clear
C)
\[54\] done
clear
D)
0 done
clear
View Solution play_arrow
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question_answer29)
The value of the integral \[\int_{-1}^{3}{(\left| x \right|+\left| x-1 \right|)dx}\] is
A)
4 done
clear
B)
9 done
clear
C)
2 done
clear
D)
\[\frac{9}{2}\] done
clear
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question_answer30)
If \[\int{\sec x\cos ec\,\,x\,\,dx=\log \left| g(x) \right|}+c,\] then what is \[g(x)\] equal to?
A)
\[\sin x\cos x\] done
clear
B)
\[{{\sec }^{2}}x\] done
clear
C)
\[\tan x\] done
clear
D)
\[\log \left| \tan x \right|\] done
clear
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question_answer31)
What is \[\int{{{\sec }^{n}}x\tan xdx}\] equal to?
A)
\[\frac{{{\sec }^{n}}x}{n}+c\] done
clear
B)
\[\frac{{{\sec }^{n-1}}x}{n-1}+c\] done
clear
C)
\[\frac{{{\tan }^{n}}x}{n}+c\] done
clear
D)
\[\frac{{{\tan }^{n-1}}x}{n-1}+c\] Where ?c? is a constant of integration. done
clear
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question_answer32)
If \[\int{{{\sin }^{3}}x{{\cos }^{5}}xdx}\]\[=A{{\sin }^{4}}x+B{{\sin }^{6}}x+C{{\sin }^{8}}x+D\] Then
A)
\[A=\frac{1}{4},B=-\frac{1}{3},C=\frac{1}{8},D\in R\] done
clear
B)
\[A=\frac{1}{8},B=\frac{1}{4},C=\frac{1}{3},D\in R\] done
clear
C)
\[A=0,B=-\frac{1}{6},C=\frac{1}{8},D\in R\] done
clear
D)
None of these done
clear
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question_answer33)
Evaluate: \[\int{\frac{1}{1+3{{\sin }^{2}}x+8{{\cos }^{2}}x}dx}\]
A)
\[\frac{1}{6}{{\tan }^{-1}}(2\tan x)+C\] done
clear
B)
\[{{\tan }^{-1}}(2\tan x)+C\] done
clear
C)
\[\frac{1}{6}{{\tan }^{-1}}\left( \frac{2\tan x}{3} \right)+C\] done
clear
D)
None of these done
clear
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question_answer34)
\[\int{{{x}^{51}}({{\tan }^{-1}}x+{{\cot }^{-1}}x)dx}\]
A)
\[\frac{{{x}^{52}}}{52}({{\tan }^{-1}}x+{{\cot }^{-1}}x)+c\] done
clear
B)
\[\frac{{{x}^{52}}}{52}({{\tan }^{-1}}x-{{\cot }^{-1}}x)+c\] done
clear
C)
\[\frac{\pi {{x}^{52}}}{104}+\frac{\pi }{2}+c\] done
clear
D)
\[\frac{{{x}^{52}}}{52}+\frac{\pi }{2}+c\] done
clear
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question_answer35)
The line \[y=\alpha \] intersects the curve \[y=g(x)\],atleast at two points. If \[\int\limits_{2}^{x}{g(t)dt=\frac{{{x}^{2}}}{2}+\int\limits_{x}^{2}{{{t}^{2}}g(t)dt}}\]then possible value of \[\alpha \] is/are-
A)
\[\left( -\frac{1}{2},\frac{1}{2} \right)\] done
clear
B)
\[\left[ -\frac{1}{2},\frac{1}{2} \right]\] done
clear
C)
\[\left( -\frac{1}{2},\frac{1}{2} \right)-\{0\}\] done
clear
D)
\[\left\{ -\frac{1}{2},0,\frac{1}{2} \right\}\] done
clear
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question_answer36)
If \[\int{f(x)\cos \,\,x\,\,dx=\frac{1}{2}{{f}^{2}}(x)+c,}\] then \[f(x)\] can be
A)
x done
clear
B)
1 done
clear
C)
\[\cos x\] done
clear
D)
\[sinx\] done
clear
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question_answer37)
The function \[f(x)=\int\limits_{-1}^{x}{t({{e}^{t}}-1)(t-1){{(t-2)}^{3}}{{(t-3)}^{5}}}\] dt has a local minimum at x =
A)
0 done
clear
B)
1, 3 done
clear
C)
2 done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer38)
What is \[\int\limits_{-\frac{\pi }{6}}^{\frac{\pi }{6}}{\frac{{{\sin }^{5}}x{{\cos }^{3}}x}{{{x}^{4}}}}dx\] equal to?
A)
\[\frac{\pi }{2}\] done
clear
B)
\[\frac{\pi }{4}\] done
clear
C)
\[\frac{\pi }{8}\] done
clear
D)
0 done
clear
View Solution play_arrow
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question_answer39)
If \[\int\limits_{-3}^{2}{f(x)dx=\frac{7}{3}}\] and \[\int\limits_{-3}^{9}{f(x)dx=-\frac{5}{6}}\], then what is the value of \[\int\limits_{2}^{9}{f(x)dx?}\]
A)
\[\frac{-19}{6}\] done
clear
B)
\[\frac{19}{6}\] done
clear
C)
\[\frac{3}{2}\] done
clear
D)
\[-\frac{3}{2}\] done
clear
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question_answer40)
If \[{{I}_{1}}=\int\limits_{0}^{\frac{\pi }{2}}{\cos (\sin \,\,x)dx;{{I}_{2}}=\int\limits_{0}^{\frac{\pi }{2}}{\sin (\cos \,\,x)dx}}\] and \[{{I}_{3}}=\int\limits_{0}^{\frac{\pi }{2}}{\cos x\,\,dx,}\] then
A)
\[{{I}_{1}}>{{I}_{3}}>{{I}_{2}}\] done
clear
B)
\[{{I}_{3}}>{{I}_{1}}>{{I}_{2}}\] done
clear
C)
\[{{I}_{1}}>{{I}_{2}}>{{I}_{3}}\] done
clear
D)
\[{{I}_{3}}>{{I}_{2}}>{{I}_{1}}\] done
clear
View Solution play_arrow
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question_answer41)
\[\int{\frac{(1+x)}{x{{(1+x{{e}^{x}})}^{2}}}dx}\] is
A)
\[\ln \,\,\left| \frac{x{{e}^{x}}}{1+x{{e}^{x}}} \right|+\frac{1}{1+x{{e}^{x}}}+C\] done
clear
B)
\[(1+x{{e}^{x}})+ln\left| \frac{x{{e}^{x}}}{1+x{{e}^{x}}} \right|+C\] done
clear
C)
\[\frac{1}{1+x{{e}^{x}}}+ln\left| x{{e}^{x}}(1+x{{e}^{x}}) \right|+C\] done
clear
D)
None of these done
clear
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question_answer42)
If \[{{u}_{n}}=\int_{0}^{\pi /4}{{{\tan }^{n}}\theta }d\theta \] then \[{{u}_{n}}+{{u}_{n-2}}\] is:
A)
\[\frac{1}{n-1}\] done
clear
B)
\[\frac{1}{n+1}\] done
clear
C)
\[\frac{1}{2n-1}\] done
clear
D)
\[\frac{1}{2n+1}\] done
clear
View Solution play_arrow
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question_answer43)
If \[f(p,q)=\int_{0}^{\pi /2}{{{\cos }^{p}}x\cos \,\,qx\,\,dx}\], then
A)
\[f(p,q)=\frac{q}{p+q}f(p-1,q-1)\] done
clear
B)
\[f(p,q)=\frac{p}{p+q}f(p-1,q-1)\] done
clear
C)
\[f(p,q)=\frac{p}{p+q}f(p-1,q-1)\] done
clear
D)
\[f(p,q)=-\frac{q}{p+q}f(p-1,q-1)\] done
clear
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question_answer44)
\[\int_{0}^{1}{[f(x)g''(x)-f''(x)g(x)]dx}\] is equal to: [Given f(0) = g (0) = 0]
A)
\[f(1)g(1)-f(1)g'(1)\] done
clear
B)
\[f(1)g'(1)+f'(1)g(1)\] done
clear
C)
\[f(1)g'(1)-f'(1)g(1)\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer45)
\[\int\limits_{0}^{\infty }{\frac{dx}{({{x}^{2}}+{{a}^{2}})({{x}^{2}}+{{b}^{2}})}}\] is
A)
\[\frac{\pi ab}{a+b}\] done
clear
B)
\[\frac{\pi }{2(a+b)}\] done
clear
C)
\[\frac{\pi }{2ab(a+b)}\] done
clear
D)
\[\frac{\pi (a+b)}{2ab}\] done
clear
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question_answer46)
If\[\int{\frac{dx}{f(x)}=\log {{\{f(x)\}}^{2}}+c}\], then what is \[f(x)\] equal to?
A)
\[2x+\alpha \] done
clear
B)
\[x+\alpha \] done
clear
C)
\[\frac{x}{2}+\alpha \] done
clear
D)
\[{{x}^{2}}+\alpha \] done
clear
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question_answer47)
What is \[\int{{{e}^{ln\,\,x}}\sin x\,\,dx}\] equal to?
A)
\[{{e}^{ln\,\,x}}(\sin \,\,x-\cos \,\,x)+c\] done
clear
B)
\[(\sin \,\,x-x\,\,\cos \,\,x)+c\] done
clear
C)
\[(x\,\,\sin \,\,x+\cos \,\,x)+c\] done
clear
D)
\[(\sin \,\,x+x\,\,\cos \,\,x)-c\] Where ?c? is a constant of integration. done
clear
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question_answer48)
\[\int{\frac{\{f(x).\phi '(x)-f'(x).\phi (x)\}}{f(x).\phi (x)}}\log \frac{f(x)}{\phi (x)}dx\] is equal to:
A)
\[\log \frac{\phi (x)}{f(x)}+k\] done
clear
B)
\[\frac{1}{2}{{\left\{ \log \frac{\phi (x)}{f(x)} \right\}}^{2}}+k\] done
clear
C)
\[\frac{\phi (x)}{f(x)}\log \frac{\phi (x)}{f(x)}+k\] done
clear
D)
None of these done
clear
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question_answer49)
\[\int\limits_{-\pi /2}^{\pi /2}{\frac{ln\,(\cos x)}{1+{{e}^{x}}.{{e}^{\sin \,\,x}}}dx=}\]
A)
\[-2\pi \,\,ln\,\,2\] done
clear
B)
\[-\frac{\pi }{4}ln\,\,2\] done
clear
C)
\[-\pi \,\,ln\,\,2\] done
clear
D)
\[-\frac{\pi }{2}ln\,\,2\] done
clear
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question_answer50)
If\[\int{f(x)dx=g(x)+c}\], then \[\int{{{f}^{-1}}(x)dx}\] is equal to
A)
\[x{{f}^{-1}}(x)+C\] done
clear
B)
\[f({{g}^{-1}})(x))+C\] done
clear
C)
\[x{{f}^{-1}}(x)-g({{f}^{-1}})(x))+C\] done
clear
D)
\[{{g}^{-1}}(x)+C\] done
clear
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question_answer51)
If\[\int{\frac{dx}{x({{x}^{n}}+1)}=A\,\,\log \left| \frac{{{x}^{n}}+1}{{{x}^{n}}} \right|+B,B\in R}\]. Then
A)
\[A=\frac{1}{2}\] done
clear
B)
\[A=-1\] done
clear
C)
\[A=-\frac{1}{n}\] done
clear
D)
\[A=\frac{1}{2n}\] done
clear
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question_answer52)
The tangent of the curve \[y=f(x)\] at the point with abscissa \[x=1\] from an angle of \[\pi /6\] and at the point \[x=2\] an angle of \[\pi /3\] and at the point \[x=3\] an angle of \[\pi /4\]. If \[f''(x)\] is continuous, then the value of \[\int\limits_{1}^{3}{f''(x)f'(x)dx+\int\limits_{2}^{3}{f''(x)dx}}\] is
A)
\[\frac{4\sqrt{3}-1}{3\sqrt{3}}\] done
clear
B)
\[\frac{3\sqrt{3}-1}{2}\] done
clear
C)
\[\frac{4-3\sqrt{3}}{3}\] done
clear
D)
None of these done
clear
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question_answer53)
If \[A=\int\limits_{0}^{1}{\frac{{{e}^{t}}}{t+1}dt,}\] then \[\int\limits_{0}^{1}{{{e}^{t}}\log (1+t)dt}\] in terms of A equals
A)
\[e\log (A)\] done
clear
B)
\[\frac{e}{2}-A\] done
clear
C)
\[e-l-\frac{A}{2}\] done
clear
D)
\[\frac{e}{2}-l-A\] done
clear
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question_answer54)
If \[\int{\frac{x{{e}^{x}}}{\sqrt{1+{{e}^{x}}}}dx=f(x)\sqrt{1+{{e}^{x}}}-2\log \,\,g(x)+C,}\] then
A)
\[f(x)=x-1\] done
clear
B)
\[g(x)=\frac{\sqrt{1+{{e}^{x}}}-1}{\sqrt{1+{{e}^{x}}}+1}\] done
clear
C)
\[g(x)=\frac{\sqrt{1+{{e}^{x}}}+1}{\sqrt{1+{{e}^{x}}}-1}\] done
clear
D)
\[f(x)=2(2-x)\] done
clear
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question_answer55)
Let \[f:(0,\infty )\to R\] and \[F(x)=\int\limits_{0}^{x}{f(t)dt}\].If \[F({{x}^{2}})={{x}^{2}}(1+x),\] then \[f(4)\] equals
A)
\[\frac{5}{4}\] done
clear
B)
7 done
clear
C)
4 done
clear
D)
2 done
clear
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question_answer56)
If \[\int{\frac{1}{1+\sin x}dx=\tan \left( \frac{x}{2}+a \right)+b}\] then
A)
\[a=-\frac{\pi }{4},b\in R\] done
clear
B)
\[a=\frac{\pi }{4},b\in R\] done
clear
C)
\[a=\frac{5\pi }{4},b\in R\] done
clear
D)
None of these done
clear
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question_answer57)
Let \[f:R\to R\] and \[g:R\to R\] be continuous functions. Then the value of \[\int\limits_{-\frac{\pi }{2}}^{\frac{\pi }{2}}{\{f(x)+f(-x)\}\{g(x)-g(-x)\}dx}\] is
A)
\[f(x)g(x)\] done
clear
B)
\[f(x)+g(x)\] done
clear
C)
0 done
clear
D)
None of theses done
clear
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question_answer58)
What is \[\int{{{\tan }^{2}}x{{\sec }^{4}}x\,dx}\] equal to?
A)
\[\frac{{{\sec }^{5}}x}{5}+\frac{{{\sec }^{3}}x}{3}+c\] done
clear
B)
\[\frac{{{\tan }^{5}}x}{5}+\frac{{{\tan }^{3}}x}{3}+c\] done
clear
C)
\[\frac{{{\tan }^{5}}x}{5}+\frac{{{\sec }^{3}}x}{3}+c\] done
clear
D)
\[\frac{{{\sec }^{5}}x}{5}+\frac{{{\tan }^{3}}x}{3}+c\] done
clear
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question_answer59)
What is \[\int\limits_{0}^{\pi /2}{\sin \,\,2x\,\,\ell n\,(\cot \,\,x)dx}\] equal to?
A)
0 done
clear
B)
\[\pi \ell n2\] done
clear
C)
\[-\pi \ell n2\] done
clear
D)
\[\frac{\pi \ell n2}{2}\] done
clear
View Solution play_arrow
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question_answer60)
\[\int{\frac{x-1}{{{(x+1)}^{2}}\sqrt{{{x}^{3}}+{{x}^{2}}+x}}dx}\] is equal to
A)
\[{{\tan }^{-1}}\sqrt{\frac{{{x}^{2}}+x+1}{x}}+C\] done
clear
B)
\[2{{\tan }^{-1}}\sqrt{\frac{{{x}^{2}}+x+1}{x}}+C\] done
clear
C)
\[3{{\tan }^{-1}}\sqrt{\frac{{{x}^{2}}+x+1}{x}}+C\] done
clear
D)
None of these done
clear
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question_answer61)
The value of \[\int\limits_{0}^{1}{\frac{dx}{{{e}^{x}}+e}}\] is equal to
A)
\[\frac{1}{e}\log \left( \frac{1+e}{2} \right)\] done
clear
B)
\[\log \left( \frac{1+e}{2} \right)\] done
clear
C)
\[\frac{1}{e}\log (1+e)\] done
clear
D)
\[\log \left( \frac{2}{1+e} \right)\] done
clear
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question_answer62)
\[\int{{{e}^{3\log x}}{{({{x}^{4}}+1)}^{-1}}dx}\] is equal to
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
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question_answer63)
\[\int{\sin 2x.\log \cos x\,\,dx}\] is equal to:
A)
\[{{\cos }^{2}}x\left( \frac{1}{2}+\log \cos x \right)+k\] done
clear
B)
\[{{\cos }^{2}}x.\log \,\,\cos \,\,x+k\] done
clear
C)
\[{{\cos }^{2}}x\left( \frac{1}{2}-\log \cos x \right)+k\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer64)
If m is an integer, then \[\int_{0}^{\pi }{\frac{\sin (2mx)}{\sin x}dx}\] is equal to:
A)
1 done
clear
B)
2 done
clear
C)
0 done
clear
D)
\[\pi \] done
clear
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question_answer65)
What is \[\int{\sin x\log (\tan x)dx}\] equal to?
A)
\[\cos x\log \tan x+\log \,\,\tan (x/2)+c\] done
clear
B)
\[-\cos x\log \tan x+\log \,\,\tan (x/2)+c\] done
clear
C)
\[\cos x\log \tan x+\log \,\,\cot \,(x/2)+c\] done
clear
D)
\[-\cos x\log \tan x+\log \,\,\cot \,(x/2)+c\] done
clear
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question_answer66)
What is \[\int{\frac{\log x}{{{(1+\log \,x)}^{2}}}dx}\] equal to?
A)
\[\frac{1}{{{\left( 1+\log x \right)}^{3}}}+c\] done
clear
B)
\[\frac{1}{{{\left( 1+\log x \right)}^{2}}}+c\] done
clear
C)
\[\frac{x}{\left( 1+\log x \right)}+c\] done
clear
D)
\[\frac{x}{{{\left( 1+\log x \right)}^{2}}}+c\] Where c is a constant. done
clear
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question_answer67)
If \[f(x)\] and \[\phi (x)\] are continuous functions on the interval \[[0,4]\] satisfying \[f(x)=f(4-x)\], \[\phi (x)+\phi (4-x)=3\] and \[\int\limits_{0}^{4}{f(x)dx=2,}\] then \[\int\limits_{0}^{4}{f(x)\phi (x)dx}\]
A)
3 done
clear
B)
6 done
clear
C)
2 done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer68)
If\[I=\int{{{\sin }^{-\frac{11}{3}}}x{{\cos }^{-\frac{1}{3}}}xdx}\]\[=A{{\cot }^{2/3}}x+B{{\cot }^{8/3}}x+C\]. Then
A)
\[A=\frac{2}{3},B=\frac{8}{3}\] done
clear
B)
\[A=-\frac{3}{2},B=-\frac{3}{8}\] done
clear
C)
\[A=\frac{3}{2},B=\frac{3}{8}\] done
clear
D)
None of these done
clear
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question_answer69)
The value of \[\int{\frac{\sin x}{\sin 4x}dx}\] is
A)
\[\frac{1}{4}\log \left| \frac{\sin x-1}{\sin x+1} \right|-\frac{1}{\sqrt{2}}\log \left| \frac{\sqrt{2}\sin x-1}{\sqrt{2}\sin x+1} \right|+C\] done
clear
B)
\[\frac{1}{8}\log \left| \frac{\cos x-1}{\cos x+1} \right|-\frac{1}{2\sqrt{2}}\log \left| \frac{\sqrt{2}\cos x-1}{\sqrt{2}\cos x+1} \right|+C\] done
clear
C)
\[\frac{1}{8}\log \left| \frac{\sin x-1}{sinx+1} \right|-\frac{1}{4\sqrt{2}}\log \left| \frac{\sqrt{2}\sin x-1}{\sqrt{2}\sin x+1} \right|+C\] done
clear
D)
None of these. done
clear
View Solution play_arrow
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question_answer70)
If \[\int{\frac{dx}{{{x}^{22}}({{x}^{7}}-6)}}\]\[=A\{In{{(p)}^{6}}+9{{p}^{2}}-2{{p}^{3}}-18p\}+c\] then
A)
\[A=\frac{1}{9072},p=\left( \frac{{{x}^{7}}-6}{{{x}^{7}}} \right)\] done
clear
B)
\[A=\frac{1}{54432},p=\left( \frac{{{x}^{7}}-6}{{{x}^{7}}} \right)\] done
clear
C)
\[A=\frac{1}{54432},p=\left( \frac{{{x}^{7}}}{{{x}^{7}}-6} \right)\] done
clear
D)
\[A=\frac{1}{9072},p={{\left( \frac{{{x}^{7}}-6}{{{x}^{7}}} \right)}^{-1}}\] done
clear
View Solution play_arrow
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question_answer71)
What is \[\int\limits_{0}^{1}{\frac{{{\tan }^{-1}}}{1+{{x}^{2}}}dx}\] equal to?
A)
\[\frac{\pi }{4}\] done
clear
B)
\[\frac{\pi }{8}\] done
clear
C)
\[\frac{{{\pi }^{2}}}{8}\] done
clear
D)
\[\frac{{{\pi }^{2}}}{32}\] done
clear
View Solution play_arrow
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question_answer72)
If \[f(x)=A\,\,\sin \left( \frac{\pi x}{2} \right)+B\] and \[f'\left( \frac{1}{2} \right)=\sqrt{2}\] and \[\int_{0}^{1}{f(x)dx=\frac{2A}{\pi }}\], then what is the value of B?
A)
\[\frac{2}{\pi }\] done
clear
B)
\[\frac{4}{\pi }\] done
clear
C)
0 done
clear
D)
1 done
clear
View Solution play_arrow
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question_answer73)
\[\int\limits_{0}^{1}{\frac{1}{\left( {{x}^{2}}+16 \right)\left( {{x}^{2}}+25 \right)}dx=}\]
A)
\[\frac{1}{5}\left[ \frac{1}{4}{{\tan }^{-1}}\left( \frac{1}{4} \right)-\frac{1}{5}{{\tan }^{-1}}\left( \frac{1}{5} \right) \right]\] done
clear
B)
\[\frac{1}{9}\left[ \frac{1}{4}{{\tan }^{-1}}\left( \frac{1}{4} \right)-\frac{1}{5}{{\tan }^{-1}}\left( \frac{1}{5} \right) \right]\] done
clear
C)
\[\frac{1}{4}\left[ \frac{1}{4}{{\tan }^{-1}}\left( \frac{1}{4} \right)-\frac{1}{5}{{\tan }^{-1}}\left( \frac{1}{5} \right) \right]\] done
clear
D)
\[\frac{1}{9}\left[ \frac{1}{5}{{\tan }^{-1}}\left( \frac{1}{4} \right)-\frac{1}{5}{{\tan }^{-1}}\left( \frac{1}{5} \right) \right]\] done
clear
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question_answer74)
If \[\int{\frac{{{x}^{2}}-x+1}{{{x}^{2}}+1}{{e}^{{{\cot }^{-1}}x}}dx=A(x){{e}^{{{\cot }^{-1}}x}}+C}\], then \[A(x)\] is equal to:
A)
\[-x\] done
clear
B)
\[x\] done
clear
C)
\[\sqrt{1-x}\] done
clear
D)
\[\sqrt{1+x}\] done
clear
View Solution play_arrow
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question_answer75)
\[\int{\frac{{{x}^{n-1}}}{{{x}^{2n}}+{{a}^{2}}}dx}=\]
A)
\[\frac{1}{na}{{\tan }^{-1}}\left( \frac{{{x}^{n}}}{a} \right)+C\] done
clear
B)
\[\frac{n}{a}{{\tan }^{-1}}\left( \frac{{{x}^{n}}}{a} \right)+C\] done
clear
C)
\[\frac{n}{a}{{\sin }^{-1}}\left( \frac{{{x}^{n}}}{a} \right)+C\] done
clear
D)
\[\frac{n}{a}{{\cos }^{-1}}\left( \frac{{{x}^{n}}}{a} \right)+C\] done
clear
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question_answer76)
\[\int\limits_{\frac{-\pi }{2}}^{\frac{\pi }{2}}{\frac{\left| x \right|dx}{8{{\cos }^{2}}2x+1}}\] has the value
A)
\[\frac{{{\pi }^{2}}}{6}\] done
clear
B)
\[\frac{{{\pi }^{2}}}{12}\] done
clear
C)
\[\frac{{{\pi }^{2}}}{24}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer77)
If \[\phi (x)=\int{{{\cot }^{4}}xdx+\frac{1}{3}{{\cot }^{3}}x-\cot x}\] and\[\phi \left( \frac{\pi }{2} \right)=\frac{\pi }{2}\] then \[\phi (x)\] is
A)
\[\pi -x\] done
clear
B)
\[x-\pi \] done
clear
C)
\[\pi /2-x\] done
clear
D)
x done
clear
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question_answer78)
If \[\int\limits_{0}^{\infty }{{{e}^{-ax}}dx=\frac{1}{a},}\] then \[\int\limits_{0}^{\infty }{{{x}^{n}}{{e}^{-ax}}dx}\] is
A)
\[\frac{{{(-1)}^{n}}n!}{{{a}^{n+1}}}\] done
clear
B)
\[\frac{{{(-1)}^{n}}(n-1)!}{{{a}^{n}}}\] done
clear
C)
\[\frac{n!}{{{a}^{n+1}}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer79)
The value of \[\int{{{e}^{ta{{n}^{-1}}}}^{x}\frac{(1+x+{{x}^{2}})}{1+{{x}^{2}}}dx}\] is
A)
\[x{{e}^{{{\tan }^{-1}}}}x+c\] done
clear
B)
\[{{\tan }^{-1}}x+C\] done
clear
C)
\[{{e}^{{{\tan }^{-1}}x}}+2x+C\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer80)
What is the value of\[\int_{0}^{1}{x{{e}^{{{x}^{2}}}}dx}\]?
A)
\[\frac{(e-1)}{2}\] done
clear
B)
\[{{e}^{2}}-1\] done
clear
C)
\[2(e-1)\] done
clear
D)
\[e-1\] done
clear
View Solution play_arrow
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question_answer81)
The value of \[\int_{0}^{\pi }{ln(1+\cos \,\,x)dx}\] is
A)
\[\frac{\pi }{2}\log 2\] done
clear
B)
\[\pi \log 2\] done
clear
C)
\[-\pi \log 2\] done
clear
D)
0 done
clear
View Solution play_arrow
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question_answer82)
If \[f(x)\] is an even function, then what is \[\int\limits_{0}^{\pi }{f(\cos x)dx}\] equal to?
A)
0 done
clear
B)
\[\int\limits_{0}^{\frac{\pi }{2}}{f(\cos x)dx}\] done
clear
C)
\[2\int\limits_{0}^{\frac{\pi }{2}}{f(\cos x)dx}\] done
clear
D)
1 done
clear
View Solution play_arrow
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question_answer83)
What is the value of\[\int_{1}^{2}{{{e}^{x}}\left( \frac{1}{x}-\frac{1}{{{x}^{2}}} \right)dx}\]?
A)
\[e\left( \frac{e}{2}-1 \right)\] done
clear
B)
\[e(e-1)\] done
clear
C)
\[e-\frac{1}{e}\] done
clear
D)
0 done
clear
View Solution play_arrow
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question_answer84)
If \[{{I}_{n}}=\int\limits_{0}^{\frac{\pi }{4}}{{{\tan }^{n}}x\,dx}\] then what is \[{{I}_{n}}+{{I}_{n-2}}\] equal to?
A)
\[\frac{1}{n}\] done
clear
B)
\[\frac{1}{(n-1)}\] done
clear
C)
\[\frac{n}{(n-1)}\] done
clear
D)
\[\frac{1}{(n-2)}\] done
clear
View Solution play_arrow
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question_answer85)
\[\int\limits_{0}^{\infty }{\left[ \frac{2}{{{e}^{x}}} \right]}\,dx\] is equal to ([x] = greatest integer \[\le \] x)
A)
\[{{\log }_{e}}2\] done
clear
B)
\[{{e}^{2}}\] done
clear
C)
0 done
clear
D)
\[\frac{2}{e}\] done
clear
View Solution play_arrow
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question_answer86)
\[I=\int{\left\{ {{\log }_{e}}{{\log }_{e}}x+\frac{1}{{{({{\log }_{e}}x)}^{2}}} \right\}dx}\] is equal to:
A)
\[x{{\log }_{e}}{{\log }_{e}}x+c\] done
clear
B)
\[x{{\log }_{e}}{{\log }_{e}}x-\frac{x}{{{\log }_{e}}x}+c\] done
clear
C)
\[x{{\log }_{e}}{{\log }_{e}}x+\frac{x}{{{\log }_{e}}x}+c\] done
clear
D)
None of these. done
clear
View Solution play_arrow
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question_answer87)
\[\int\limits_{0}^{2\pi }{\log \left( \frac{a+b\sec x}{a-b\sec x} \right)}dx=\]
A)
0 done
clear
B)
\[\pi /2\] done
clear
C)
\[\frac{\pi (a+b)}{a-b}\] done
clear
D)
\[\frac{\pi }{2}({{a}^{2}}-{{b}^{2}})\] done
clear
View Solution play_arrow
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question_answer88)
If\[\int\limits_{1}^{2}{\left\{ {{K}^{2}}+(4-4K)x+4{{x}^{3}} \right\}dx\le 12}\], then which one of the following is correct?
A)
\[K=3\] done
clear
B)
\[0\le K<3\] done
clear
C)
\[K\le 4\] done
clear
D)
\[K=0\] done
clear
View Solution play_arrow
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question_answer89)
If \[f(x)=a+bx+c{{x}^{2}},\] then what is \[\int_{0}^{1}{f(x)dx}\] equal to?
A)
\[[f(0)+4f(1/2)+f(1)]/6\] done
clear
B)
\[[f(0)+4f(1/2)+f(1)]/3\] done
clear
C)
\[[f(0)+4f(1/2)+f(1)]\] done
clear
D)
\[[f(0)+2f(1/2)+f(1)]/6\] done
clear
View Solution play_arrow
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question_answer90)
Let \[F(x)=f(x)+f\left( \frac{1}{x} \right),\] where \[f(x)=\int\limits_{l}^{x}{\frac{\log t}{1+t}dt}\], Then \[F(e)\] equals
A)
1 done
clear
B)
2 done
clear
C)
1/2 done
clear
D)
0 done
clear
View Solution play_arrow