JEE Main & Advanced Mathematics Indefinite Integrals Question Bank

Self Evaluation Test - Integrals Total Questions - 90

1)

\[\int{\frac{\sqrt{x}}{1+\sqrt[4]{{{x}^{3}}}}}dx\] is equal to

A)
\[\frac{4}{3}\left[ 1+{{x}^{3/4}}+\log (1+{{x}^{3/4}}) \right]+C\]

B)
\[\frac{4}{3}\left[ 1+{{x}^{3/4}}-\log (1+{{x}^{3/4}}) \right]+C\]

C)
\[\frac{4}{3}\left[ 1-{{x}^{3/4}}+\log (1+{{x}^{3/4}}) \right]+C\]

D)
None of these
View Solution
2)

\[\int{\frac{({{x}^{2}}-1)}{x\sqrt{{{x}^{4}}+3{{x}^{2}}+1}}dx}\] is equal to

A)
\[\log \left| x+\frac{1}{x}+\sqrt{{{x}^{2}}+\frac{1}{{{x}^{2}}}+3} \right|+C\]

B)
\[\log \left| x-\frac{1}{x}+\sqrt{{{x}^{2}}+\frac{1}{{{x}^{2}}}-3} \right|+C\]

C)
\[\log \left| x+\sqrt{{{x}^{2}}+3} \right|+C\]

D)
None of these
View Solution
3)

\[\int{\frac{(1+x){{e}^{x}}}{\cot (x{{e}^{x}})}dx}\] is equal to

A)
\[\log \left| \cos (x{{e}^{x}}) \right|+C\]

B)
\[\log \left| \cot (x{{e}^{x}}) \right|+C\]

C)
\[\log \left| sec(x{{e}^{-x}}) \right|+C\]

D)
\[\log \left| sec(x{{e}^{x}}) \right|+C\]
View Solution
4)

\[\int{\frac{dx}{\cos x+\sqrt{3}\sin x}}\] equals

A)
\[\log \tan \left( \frac{x}{2}+\frac{\pi }{12} \right)+C\]

B)
\[\log \tan \left( \frac{x}{2}-\frac{\pi }{12} \right)+C\]

C)
\[\frac{1}{2}\log \tan \left( \frac{x}{2}+\frac{\pi }{12} \right)+C\]

D)
\[\frac{1}{2}\log \tan \left( \frac{x}{2}-\frac{\pi }{12} \right)+C\]
View Solution
5)

If \[I=\int{\frac{1}{2p}\sqrt{\frac{p-1}{p+1}}dp=f(p)+c}\], then f(p) is equal to:

A)
\[\frac{1}{2}\ell n\left[ p-\sqrt{{{p}^{2}}-1} \right]\]

B)
\[\frac{1}{2}{{\cos }^{-1}}p+\frac{1}{2}{{\sec }^{-1}}p\]

C)
\[\ell n\sqrt{p+\sqrt{{{p}^{2}}-1}}-\frac{1}{2}{{\sec }^{-1}}p\]

D)
None of the above.
View Solution
6)

Let\[f(x)=\int{{{e}^{x}}(x-1)(x-2)}dx\]. Then f decreases in the interval

A)
\[(-\infty ,-2)\]

B)
\[(-2,-1)\]

C)
\[(1,\,\,2)\]

D)
\[(2,+\infty )\]
View Solution
7)

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)
\[8f'(1)\]

B)
\[4f'(1)\]

C)
\[2f'(1)\]

D)
\[f'(1)\]
View Solution
8)

\[\int{{{\left( x+\frac{1}{x} \right)}^{n+5}}\left( \frac{{{x}^{2}}-1}{{{x}^{2}}} \right)dx}\] is equal to:

A)
\[\frac{{{\left( x+\frac{1}{x} \right)}^{n+6}}}{n+6}+c\]

B)
\[{{\left[ \frac{{{x}^{2}}+1}{{{x}^{2}}} \right]}^{n+6}}(n+6)+c\]

C)
\[{{\left[ \frac{x}{{{x}^{2}}+1} \right]}^{n+6}}(n+6)+c\]

D)
None of these
View Solution
9)

Find the value of \[\int\limits_{0}^{9}{[\sqrt{x}+2]dx}\] where \[[\cdot ]\] is the greatest integer function:

A)
31

B)
22

C)
23

D)
None of these
View Solution
10)

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)
\[{{a}^{2}}{{\sin }^{2}}x+{{b}^{2}}{{\cos }^{2}}x+C\]

B)
\[{{a}^{2}}{{\sin }^{2}}x-{{b}^{2}}{{\cos }^{2}}x+C\]

C)
\[{{a}^{2}}{{\cos }^{2}}x+{{b}^{2}}si{{n}^{2}}x+C\]

D)
\[{{a}^{2}}{{\cos }^{2}}x-{{b}^{2}}si{{n}^{2}}x+C\]
View Solution
11)

\[\int{\frac{{{x}^{2}}}{({{x}^{2}}+1)({{x}^{2}}+4)}dx}\] is equal to

A)
\[{{\tan }^{-1}}x+2{{\tan }^{-1}}\left( \frac{x}{2} \right)+C\]

B)
\[{{\tan }^{-1}}\left( \frac{x}{2} \right)-4{{\tan }^{-1}}x+C\]

C)
\[-\frac{1}{3}{{\tan }^{-1}}x+\frac{2}{3}{{\tan }^{-1}}\left( \frac{x}{2} \right)+C\]

D)
\[4{{\tan }^{-1}}\left( \frac{x}{2} \right)-2{{\tan }^{-1}}x+C\]
View Solution
12)

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

B)
\[-3\]

C)
\[-1\]

D)
\[2\]
View Solution
13)

\[\int{\sqrt{\frac{x}{1-x}}dx}\] is equal to

A)
\[{{\sin }^{-1}}\sqrt{x}+c\]

B)
\[{{\sin }^{-1}}\{\sqrt{x}-\sqrt{x(1-x)\}}+c\]

C)
\[{{\sin }^{-1}}\sqrt{x(1-x)}+c\]

D)
\[{{\sin }^{-1}}\sqrt{x}-\sqrt{x(1-x)}+c\]
View Solution
14)

What is \[\int{\frac{{{e}^{x}}(1+x)}{{{\cos }^{2}}\left( x{{e}^{x}} \right)}dx}\] equal to?

A)
\[x{{e}^{x}}+c\]

B)
\[\cos (x{{e}^{x}})+c\]

C)
\[\tan (x{{e}^{x}})+c\]

D)
\[x\cos ec(x{{e}^{x}})+c\] Where c is a constant of integration.
View Solution
15)

If\[f(x)=ln(x-\sqrt{1+{{x}^{2}}})\], then what is \[\int{f''(x)dx}\] equal to?

A)
\[\frac{1}{(x-\sqrt{1+{{x}^{2}}})}+c\]

B)
\[-\frac{1}{\sqrt{1+{{x}^{2}}}}+c\]

C)
\[-\sqrt{1+{{x}^{2}}}+c\]

D)
\[\text{ln}\,(x-\sqrt{1+{{x}^{2}}})+c\]
View Solution
16)

What is the value of\[\int\limits_{0}^{1}{(x-1){{e}^{-x}}dx}\]?

A)
0

B)
e

C)
\[\frac{1}{e}\]

D)
\[\frac{-1}{e}\]
View Solution
17)

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)
\[\pi \]

B)
\[\frac{\pi }{2}\]

C)
\[\frac{\pi }{4}\]

D)
1
View Solution
18)

\[\underset{n\to \infty }{\mathop{Lim}}\,{{\left\{ \frac{n!}{{{(kn)}^{n}}} \right\}}^{\frac{1}{n}}},\] where \[k\ne 0\] is a constant and \[n\in N\] is equal to

A)
ke

B)
\[{{k}^{-1}}e\]

C)
\[k{{e}^{-1}}\]

D)
\[{{k}^{-1}}{{e}^{-1}}\]
View Solution
19)

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)
\[{{I}_{1}}=2{{I}_{2}}\]

B)
\[2{{I}_{1}}={{I}_{2}}\]

C)
\[{{I}_{1}}={{I}_{2}}\]

D)
\[{{I}_{1}}+{{I}_{2}}=0\]
View Solution
20)

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)
\[{{l}^{r+1}}(x)+C\]

B)
\[\frac{{{l}^{r+1}}(x)}{r+1}+C\]

C)
\[{{l}^{r}}(x)+C\]

D)
None
View Solution
21)

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)
\[x-{{\sin }^{-1}}x\]

B)
\[{{\sin }^{-1}}x-x\]

C)
\[x+{{\sin }^{-1}}x\]

D)
\[{{\sin }^{-1}}x-{{x}^{2}}\]
View Solution
22)

\[{{A}_{n}}=\int\limits_{0}^{\pi /2}{\frac{\sin (2n-1)x}{\sin x}}dx;{{B}_{n}}=\int\limits_{0}^{\pi /2}{{{\left( \frac{\sin nx}{\sin x} \right)}^{2}}dx;}\] For \[n\in N,\] then

A)
\[{{A}_{n+1}}={{A}_{n}},{{B}_{n+1}}-{{B}_{n}}={{A}_{n+1}}\]

B)
\[{{B}_{n+1}}={{B}_{n}}\]

C)
\[{{A}_{n+1}}-{{A}_{n}}={{B}_{n+1}}\]

D)
None of these
View Solution
23)

If \[\int{x\log \left( 1+\frac{1}{x} \right)dx}\]\[=f(x)\log (x+1)+g(x){{x}^{2}}+Lx+C\], then

A)
\[f(x)=\frac{1}{2}{{x}^{2}}\]

B)
\[g(x)=\log x\]

C)
\[L=1\]

D)
None of these
View Solution
24)

If \[\int{g(x)dx=g(x),}\] then\[\int{g(x)\{f(x)+f'(x)\}dx}\] is equal to

A)
\[g(x)f(x)-g(x)f'(x)+C\]

B)
\[g(x)f'(x)+C\]

C)
\[g(x)f(x)+C\]

D)
\[g(x){{f}^{2}}(x)+C\]
View Solution
25)

\[\int{\frac{dx}{\sin x(3+{{\cos }^{2}}x)}}\] is equal to

A)
\[\log \left| {{y}^{2}}-1 \right|-{{\tan }^{-1}}y+C\]

B)
\[{{\tan }^{-1}}\frac{y}{\sqrt{3}}+C\]

C)
\[\log \left| \frac{y-1}{y+1} \right|+C\]

D)
\[\frac{1}{4}\log \left| \frac{y-1}{y+1} \right|-\frac{1}{4\sqrt{3}}{{\tan }^{-1}}\frac{y}{\sqrt{3}}+C\]
View Solution
26)

\[\int\limits_{0}^{\pi }{xf(\sin \,\,x)dx}\] is equal to

A)
\[\pi \int\limits_{0}^{\pi }{f(cos\,\,x)dx}\]

B)
\[\pi \int\limits_{0}^{\pi }{f(sin\,\,x)dx}\]

C)
\[\frac{\pi }{2}\int\limits_{0}^{\pi /2}{f(sin\,\,x)dx}\]

D)
\[\pi \int\limits_{0}^{\pi /2}{f(cos\,\,x)dx}\]
View Solution
27)

\[\int{32{{x}^{3}}{{(\log \,\,x)}^{2}}dx}\] is equal to:

A)
\[8{{x}^{4}}{{(\log \,\,x)}^{2}}+C\]

B)
\[{{x}^{4}}\{8{{(\log \,\,x)}^{2}}-4(\log \,\,x)+1\}+C\]

C)
\[{{x}^{4}}\{8{{(\log \,\,x)}^{2}}-4(\log \,\,x)\}+C\]

D)
\[{{x}^{3}}\{{{(\log \,\,x)}^{2}}-2\log \,\,x\}+C\]
View Solution
28)

\[\left[ \sum\limits_{n=1}^{10}{\int\limits_{-2n-1}^{-2n}{{{\sin }^{27}}xdx}} \right]+\left[ \sum\limits_{n=1}^{10}{\int\limits_{2n}^{2n+1}{{{\sin }^{27}}}xdx} \right]=\]

A)
\[{{27}^{2}}\]

B)
\[-54\]

C)
\[54\]

D)
0
View Solution
29)

The value of the integral \[\int_{-1}^{3}{(\left| x \right|+\left| x-1 \right|)dx}\] is

A)
4

B)
9

C)
2

D)
\[\frac{9}{2}\]
View Solution
30)

If \[\int{\sec x\cos ec\,\,x\,\,dx=\log \left| g(x) \right|}+c,\] then what is \[g(x)\] equal to?

A)
\[\sin x\cos x\]

B)
\[{{\sec }^{2}}x\]

C)
\[\tan x\]

D)
\[\log \left| \tan x \right|\]
View Solution
31)

What is \[\int{{{\sec }^{n}}x\tan xdx}\] equal to?

A)
\[\frac{{{\sec }^{n}}x}{n}+c\]

B)
\[\frac{{{\sec }^{n-1}}x}{n-1}+c\]

C)
\[\frac{{{\tan }^{n}}x}{n}+c\]

D)
\[\frac{{{\tan }^{n-1}}x}{n-1}+c\] Where ?c? is a constant of integration.
View Solution
32)

If \[\int{{{\sin }^{3}}x{{\cos }^{5}}xdx}\]\[=A{{\sin }^{4}}x+B{{\sin }^{6}}x+C{{\sin }^{8}}x+D\] Then

A)
\[A=\frac{1}{4},B=-\frac{1}{3},C=\frac{1}{8},D\in R\]

B)
\[A=\frac{1}{8},B=\frac{1}{4},C=\frac{1}{3},D\in R\]

C)
\[A=0,B=-\frac{1}{6},C=\frac{1}{8},D\in R\]

D)
None of these
View Solution
33)

Evaluate: \[\int{\frac{1}{1+3{{\sin }^{2}}x+8{{\cos }^{2}}x}dx}\]

A)
\[\frac{1}{6}{{\tan }^{-1}}(2\tan x)+C\]

B)
\[{{\tan }^{-1}}(2\tan x)+C\]

C)
\[\frac{1}{6}{{\tan }^{-1}}\left( \frac{2\tan x}{3} \right)+C\]

D)
None of these
View Solution
34)

\[\int{{{x}^{51}}({{\tan }^{-1}}x+{{\cot }^{-1}}x)dx}\]

A)
\[\frac{{{x}^{52}}}{52}({{\tan }^{-1}}x+{{\cot }^{-1}}x)+c\]

B)
\[\frac{{{x}^{52}}}{52}({{\tan }^{-1}}x-{{\cot }^{-1}}x)+c\]

C)
\[\frac{\pi {{x}^{52}}}{104}+\frac{\pi }{2}+c\]

D)
\[\frac{{{x}^{52}}}{52}+\frac{\pi }{2}+c\]
View Solution
35)

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)
\[\left( -\frac{1}{2},\frac{1}{2} \right)\]

B)
\[\left[ -\frac{1}{2},\frac{1}{2} \right]\]

C)
\[\left( -\frac{1}{2},\frac{1}{2} \right)-\{0\}\]

D)
\[\left\{ -\frac{1}{2},0,\frac{1}{2} \right\}\]
View Solution
36)

If \[\int{f(x)\cos \,\,x\,\,dx=\frac{1}{2}{{f}^{2}}(x)+c,}\] then \[f(x)\] can be

A)
x

B)
1

C)
\[\cos x\]

D)
\[sinx\]
View Solution
37)

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

B)
1, 3

C)
2

D)
None of these
View Solution
38)

What is \[\int\limits_{-\frac{\pi }{6}}^{\frac{\pi }{6}}{\frac{{{\sin }^{5}}x{{\cos }^{3}}x}{{{x}^{4}}}}dx\] equal to?

A)
\[\frac{\pi }{2}\]

B)
\[\frac{\pi }{4}\]

C)
\[\frac{\pi }{8}\]

D)
0
View Solution
39)

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)
\[\frac{-19}{6}\]

B)
\[\frac{19}{6}\]

C)
\[\frac{3}{2}\]

D)
\[-\frac{3}{2}\]
View Solution
40)

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)
\[{{I}_{1}}>{{I}_{3}}>{{I}_{2}}\]

B)
\[{{I}_{3}}>{{I}_{1}}>{{I}_{2}}\]

C)
\[{{I}_{1}}>{{I}_{2}}>{{I}_{3}}\]

D)
\[{{I}_{3}}>{{I}_{2}}>{{I}_{1}}\]
View Solution
41)

\[\int{\frac{(1+x)}{x{{(1+x{{e}^{x}})}^{2}}}dx}\] is

A)
\[\ln \,\,\left| \frac{x{{e}^{x}}}{1+x{{e}^{x}}} \right|+\frac{1}{1+x{{e}^{x}}}+C\]

B)
\[(1+x{{e}^{x}})+ln\left| \frac{x{{e}^{x}}}{1+x{{e}^{x}}} \right|+C\]

C)
\[\frac{1}{1+x{{e}^{x}}}+ln\left| x{{e}^{x}}(1+x{{e}^{x}}) \right|+C\]

D)
None of these
View Solution
42)

If \[{{u}_{n}}=\int_{0}^{\pi /4}{{{\tan }^{n}}\theta }d\theta \] then \[{{u}_{n}}+{{u}_{n-2}}\] is:

A)
\[\frac{1}{n-1}\]

B)
\[\frac{1}{n+1}\]

C)
\[\frac{1}{2n-1}\]

D)
\[\frac{1}{2n+1}\]
View Solution
43)

If \[f(p,q)=\int_{0}^{\pi /2}{{{\cos }^{p}}x\cos \,\,qx\,\,dx}\], then

A)
\[f(p,q)=\frac{q}{p+q}f(p-1,q-1)\]

B)
\[f(p,q)=\frac{p}{p+q}f(p-1,q-1)\]

C)
\[f(p,q)=\frac{p}{p+q}f(p-1,q-1)\]

D)
\[f(p,q)=-\frac{q}{p+q}f(p-1,q-1)\]
View Solution
44)

\[\int_{0}^{1}{[f(x)g''(x)-f''(x)g(x)]dx}\] is equal to: [Given f(0) = g (0) = 0]

A)
\[f(1)g(1)-f(1)g'(1)\]

B)
\[f(1)g'(1)+f'(1)g(1)\]

C)
\[f(1)g'(1)-f'(1)g(1)\]

D)
None of these
View Solution
45)

\[\int\limits_{0}^{\infty }{\frac{dx}{({{x}^{2}}+{{a}^{2}})({{x}^{2}}+{{b}^{2}})}}\] is

A)
\[\frac{\pi ab}{a+b}\]

B)
\[\frac{\pi }{2(a+b)}\]

C)
\[\frac{\pi }{2ab(a+b)}\]

D)
\[\frac{\pi (a+b)}{2ab}\]
View Solution
46)

If\[\int{\frac{dx}{f(x)}=\log {{\{f(x)\}}^{2}}+c}\], then what is \[f(x)\] equal to?

A)
\[2x+\alpha \]

B)
\[x+\alpha \]

C)
\[\frac{x}{2}+\alpha \]

D)
\[{{x}^{2}}+\alpha \]
View Solution
47)

What is \[\int{{{e}^{ln\,\,x}}\sin x\,\,dx}\] equal to?

A)
\[{{e}^{ln\,\,x}}(\sin \,\,x-\cos \,\,x)+c\]

B)
\[(\sin \,\,x-x\,\,\cos \,\,x)+c\]

C)
\[(x\,\,\sin \,\,x+\cos \,\,x)+c\]

D)
\[(\sin \,\,x+x\,\,\cos \,\,x)-c\] Where ?c? is a constant of integration.
View Solution
48)

\[\int{\frac{\{f(x).\phi '(x)-f'(x).\phi (x)\}}{f(x).\phi (x)}}\log \frac{f(x)}{\phi (x)}dx\] is equal to:

A)
\[\log \frac{\phi (x)}{f(x)}+k\]

B)
\[\frac{1}{2}{{\left\{ \log \frac{\phi (x)}{f(x)} \right\}}^{2}}+k\]

C)
\[\frac{\phi (x)}{f(x)}\log \frac{\phi (x)}{f(x)}+k\]

D)
None of these
View Solution
49)

\[\int\limits_{-\pi /2}^{\pi /2}{\frac{ln\,(\cos x)}{1+{{e}^{x}}.{{e}^{\sin \,\,x}}}dx=}\]

A)
\[-2\pi \,\,ln\,\,2\]

B)
\[-\frac{\pi }{4}ln\,\,2\]

C)
\[-\pi \,\,ln\,\,2\]

D)
\[-\frac{\pi }{2}ln\,\,2\]
View Solution
50)

If\[\int{f(x)dx=g(x)+c}\], then \[\int{{{f}^{-1}}(x)dx}\] is equal to

A)
\[x{{f}^{-1}}(x)+C\]

B)
\[f({{g}^{-1}})(x))+C\]

C)
\[x{{f}^{-1}}(x)-g({{f}^{-1}})(x))+C\]

D)
\[{{g}^{-1}}(x)+C\]
View Solution
51)

If\[\int{\frac{dx}{x({{x}^{n}}+1)}=A\,\,\log \left| \frac{{{x}^{n}}+1}{{{x}^{n}}} \right|+B,B\in R}\]. Then

A)
\[A=\frac{1}{2}\]

B)
\[A=-1\]

C)
\[A=-\frac{1}{n}\]

D)
\[A=\frac{1}{2n}\]
View Solution
52)

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)
\[\frac{4\sqrt{3}-1}{3\sqrt{3}}\]

B)
\[\frac{3\sqrt{3}-1}{2}\]

C)
\[\frac{4-3\sqrt{3}}{3}\]

D)
None of these
View Solution
53)

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)
\[e\log (A)\]

B)
\[\frac{e}{2}-A\]

C)
\[e-l-\frac{A}{2}\]

D)
\[\frac{e}{2}-l-A\]
View Solution
54)

If \[\int{\frac{x{{e}^{x}}}{\sqrt{1+{{e}^{x}}}}dx=f(x)\sqrt{1+{{e}^{x}}}-2\log \,\,g(x)+C,}\] then

A)
\[f(x)=x-1\]

B)
\[g(x)=\frac{\sqrt{1+{{e}^{x}}}-1}{\sqrt{1+{{e}^{x}}}+1}\]

C)
\[g(x)=\frac{\sqrt{1+{{e}^{x}}}+1}{\sqrt{1+{{e}^{x}}}-1}\]

D)
\[f(x)=2(2-x)\]
View Solution
55)

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)
\[\frac{5}{4}\]

B)
7

C)
4

D)
2
View Solution
56)

If \[\int{\frac{1}{1+\sin x}dx=\tan \left( \frac{x}{2}+a \right)+b}\] then

A)
\[a=-\frac{\pi }{4},b\in R\]

B)
\[a=\frac{\pi }{4},b\in R\]

C)
\[a=\frac{5\pi }{4},b\in R\]

D)
None of these
View Solution
57)

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)
\[f(x)g(x)\]

B)
\[f(x)+g(x)\]

C)
0

D)
None of theses
View Solution
58)

What is \[\int{{{\tan }^{2}}x{{\sec }^{4}}x\,dx}\] equal to?

A)
\[\frac{{{\sec }^{5}}x}{5}+\frac{{{\sec }^{3}}x}{3}+c\]

B)
\[\frac{{{\tan }^{5}}x}{5}+\frac{{{\tan }^{3}}x}{3}+c\]

C)
\[\frac{{{\tan }^{5}}x}{5}+\frac{{{\sec }^{3}}x}{3}+c\]

D)
\[\frac{{{\sec }^{5}}x}{5}+\frac{{{\tan }^{3}}x}{3}+c\]
View Solution
59)

What is \[\int\limits_{0}^{\pi /2}{\sin \,\,2x\,\,\ell n\,(\cot \,\,x)dx}\] equal to?

A)
0

B)
\[\pi \ell n2\]

C)
\[-\pi \ell n2\]

D)
\[\frac{\pi \ell n2}{2}\]
View Solution
60)

\[\int{\frac{x-1}{{{(x+1)}^{2}}\sqrt{{{x}^{3}}+{{x}^{2}}+x}}dx}\] is equal to

A)
\[{{\tan }^{-1}}\sqrt{\frac{{{x}^{2}}+x+1}{x}}+C\]

B)
\[2{{\tan }^{-1}}\sqrt{\frac{{{x}^{2}}+x+1}{x}}+C\]

C)
\[3{{\tan }^{-1}}\sqrt{\frac{{{x}^{2}}+x+1}{x}}+C\]

D)
None of these
View Solution
61)

The value of \[\int\limits_{0}^{1}{\frac{dx}{{{e}^{x}}+e}}\] is equal to

A)
\[\frac{1}{e}\log \left( \frac{1+e}{2} \right)\]

B)
\[\log \left( \frac{1+e}{2} \right)\]

C)
\[\frac{1}{e}\log (1+e)\]

D)
\[\log \left( \frac{2}{1+e} \right)\]
View Solution
62)

\[\int{{{e}^{3\log x}}{{({{x}^{4}}+1)}^{-1}}dx}\] is equal to

A)
\[\log ({{x}^{4}}+1)+C\]

B)
\[\frac{1}{4}\log ({{x}^{4}}+1)+C\]

C)
\[-\log ({{x}^{4}}+1)+C\]

D)
None of these
View Solution
63)

\[\int{\sin 2x.\log \cos x\,\,dx}\] is equal to:

A)
\[{{\cos }^{2}}x\left( \frac{1}{2}+\log \cos x \right)+k\]

B)
\[{{\cos }^{2}}x.\log \,\,\cos \,\,x+k\]

C)
\[{{\cos }^{2}}x\left( \frac{1}{2}-\log \cos x \right)+k\]

D)
None of these
View Solution
64)

If m is an integer, then \[\int_{0}^{\pi }{\frac{\sin (2mx)}{\sin x}dx}\] is equal to:

A)
1

B)
2

C)
0

D)
\[\pi \]
View Solution
65)

What is \[\int{\sin x\log (\tan x)dx}\] equal to?

A)
\[\cos x\log \tan x+\log \,\,\tan (x/2)+c\]

B)
\[-\cos x\log \tan x+\log \,\,\tan (x/2)+c\]

C)
\[\cos x\log \tan x+\log \,\,\cot \,(x/2)+c\]

D)
\[-\cos x\log \tan x+\log \,\,\cot \,(x/2)+c\]
View Solution
66)

What is \[\int{\frac{\log x}{{{(1+\log \,x)}^{2}}}dx}\] equal to?

A)
\[\frac{1}{{{\left( 1+\log x \right)}^{3}}}+c\]

B)
\[\frac{1}{{{\left( 1+\log x \right)}^{2}}}+c\]

C)
\[\frac{x}{\left( 1+\log x \right)}+c\]

D)
\[\frac{x}{{{\left( 1+\log x \right)}^{2}}}+c\] Where c is a constant.
View Solution
67)

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

B)
6

C)
2

D)
None of these
View Solution
68)

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)
\[A=\frac{2}{3},B=\frac{8}{3}\]

B)
\[A=-\frac{3}{2},B=-\frac{3}{8}\]

C)
\[A=\frac{3}{2},B=\frac{3}{8}\]

D)
None of these
View Solution
69)

The value of \[\int{\frac{\sin x}{\sin 4x}dx}\] is

A)
\[\frac{1}{4}\log \left| \frac{\sin x-1}{\sin x+1} \right|-\frac{1}{\sqrt{2}}\log \left| \frac{\sqrt{2}\sin x-1}{\sqrt{2}\sin x+1} \right|+C\]

B)
\[\frac{1}{8}\log \left| \frac{\cos x-1}{\cos x+1} \right|-\frac{1}{2\sqrt{2}}\log \left| \frac{\sqrt{2}\cos x-1}{\sqrt{2}\cos x+1} \right|+C\]

C)
\[\frac{1}{8}\log \left| \frac{\sin x-1}{sinx+1} \right|-\frac{1}{4\sqrt{2}}\log \left| \frac{\sqrt{2}\sin x-1}{\sqrt{2}\sin x+1} \right|+C\]

D)
None of these.
View Solution
70)

If \[\int{\frac{dx}{{{x}^{22}}({{x}^{7}}-6)}}\]\[=A\{In{{(p)}^{6}}+9{{p}^{2}}-2{{p}^{3}}-18p\}+c\] then

A)
\[A=\frac{1}{9072},p=\left( \frac{{{x}^{7}}-6}{{{x}^{7}}} \right)\]

B)
\[A=\frac{1}{54432},p=\left( \frac{{{x}^{7}}-6}{{{x}^{7}}} \right)\]

C)
\[A=\frac{1}{54432},p=\left( \frac{{{x}^{7}}}{{{x}^{7}}-6} \right)\]

D)
\[A=\frac{1}{9072},p={{\left( \frac{{{x}^{7}}-6}{{{x}^{7}}} \right)}^{-1}}\]
View Solution
71)

What is \[\int\limits_{0}^{1}{\frac{{{\tan }^{-1}}}{1+{{x}^{2}}}dx}\] equal to?

A)
\[\frac{\pi }{4}\]

B)
\[\frac{\pi }{8}\]

C)
\[\frac{{{\pi }^{2}}}{8}\]

D)
\[\frac{{{\pi }^{2}}}{32}\]
View Solution
72)

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)
\[\frac{2}{\pi }\]

B)
\[\frac{4}{\pi }\]

C)
0

D)
1
View Solution
73)

\[\int\limits_{0}^{1}{\frac{1}{\left( {{x}^{2}}+16 \right)\left( {{x}^{2}}+25 \right)}dx=}\]

A)
\[\frac{1}{5}\left[ \frac{1}{4}{{\tan }^{-1}}\left( \frac{1}{4} \right)-\frac{1}{5}{{\tan }^{-1}}\left( \frac{1}{5} \right) \right]\]

B)
\[\frac{1}{9}\left[ \frac{1}{4}{{\tan }^{-1}}\left( \frac{1}{4} \right)-\frac{1}{5}{{\tan }^{-1}}\left( \frac{1}{5} \right) \right]\]

C)
\[\frac{1}{4}\left[ \frac{1}{4}{{\tan }^{-1}}\left( \frac{1}{4} \right)-\frac{1}{5}{{\tan }^{-1}}\left( \frac{1}{5} \right) \right]\]

D)
\[\frac{1}{9}\left[ \frac{1}{5}{{\tan }^{-1}}\left( \frac{1}{4} \right)-\frac{1}{5}{{\tan }^{-1}}\left( \frac{1}{5} \right) \right]\]
View Solution
74)

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)
\[-x\]

B)
\[x\]

C)
\[\sqrt{1-x}\]

D)
\[\sqrt{1+x}\]
View Solution
75)

\[\int{\frac{{{x}^{n-1}}}{{{x}^{2n}}+{{a}^{2}}}dx}=\]

A)
\[\frac{1}{na}{{\tan }^{-1}}\left( \frac{{{x}^{n}}}{a} \right)+C\]

B)
\[\frac{n}{a}{{\tan }^{-1}}\left( \frac{{{x}^{n}}}{a} \right)+C\]

C)
\[\frac{n}{a}{{\sin }^{-1}}\left( \frac{{{x}^{n}}}{a} \right)+C\]

D)
\[\frac{n}{a}{{\cos }^{-1}}\left( \frac{{{x}^{n}}}{a} \right)+C\]
View Solution
76)

\[\int\limits_{\frac{-\pi }{2}}^{\frac{\pi }{2}}{\frac{\left| x \right|dx}{8{{\cos }^{2}}2x+1}}\] has the value

A)
\[\frac{{{\pi }^{2}}}{6}\]

B)
\[\frac{{{\pi }^{2}}}{12}\]

C)
\[\frac{{{\pi }^{2}}}{24}\]

D)
None of these
View Solution
77)

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)
\[\pi -x\]

B)
\[x-\pi \]

C)
\[\pi /2-x\]

D)
x
View Solution
78)

If \[\int\limits_{0}^{\infty }{{{e}^{-ax}}dx=\frac{1}{a},}\] then \[\int\limits_{0}^{\infty }{{{x}^{n}}{{e}^{-ax}}dx}\] is

A)
\[\frac{{{(-1)}^{n}}n!}{{{a}^{n+1}}}\]

B)
\[\frac{{{(-1)}^{n}}(n-1)!}{{{a}^{n}}}\]

C)
\[\frac{n!}{{{a}^{n+1}}}\]

D)
None of these
View Solution
79)

The value of \[\int{{{e}^{ta{{n}^{-1}}}}^{x}\frac{(1+x+{{x}^{2}})}{1+{{x}^{2}}}dx}\] is

A)
\[x{{e}^{{{\tan }^{-1}}}}x+c\]

B)
\[{{\tan }^{-1}}x+C\]

C)
\[{{e}^{{{\tan }^{-1}}x}}+2x+C\]

D)
None of these
View Solution
80)

What is the value of\[\int_{0}^{1}{x{{e}^{{{x}^{2}}}}dx}\]?

A)
\[\frac{(e-1)}{2}\]

B)
\[{{e}^{2}}-1\]

C)
\[2(e-1)\]

D)
\[e-1\]
View Solution
81)

The value of \[\int_{0}^{\pi }{ln(1+\cos \,\,x)dx}\] is

A)
\[\frac{\pi }{2}\log 2\]

B)
\[\pi \log 2\]

C)
\[-\pi \log 2\]

D)
0
View Solution
82)

If \[f(x)\] is an even function, then what is \[\int\limits_{0}^{\pi }{f(\cos x)dx}\] equal to?

A)
0

B)
\[\int\limits_{0}^{\frac{\pi }{2}}{f(\cos x)dx}\]

C)
\[2\int\limits_{0}^{\frac{\pi }{2}}{f(\cos x)dx}\]

D)
1
View Solution
83)

What is the value of\[\int_{1}^{2}{{{e}^{x}}\left( \frac{1}{x}-\frac{1}{{{x}^{2}}} \right)dx}\]?

A)
\[e\left( \frac{e}{2}-1 \right)\]

B)
\[e(e-1)\]

C)
\[e-\frac{1}{e}\]

D)
0
View Solution
84)

If \[{{I}_{n}}=\int\limits_{0}^{\frac{\pi }{4}}{{{\tan }^{n}}x\,dx}\] then what is \[{{I}_{n}}+{{I}_{n-2}}\] equal to?

A)
\[\frac{1}{n}\]

B)
\[\frac{1}{(n-1)}\]

C)
\[\frac{n}{(n-1)}\]

D)
\[\frac{1}{(n-2)}\]
View Solution
85)

\[\int\limits_{0}^{\infty }{\left[ \frac{2}{{{e}^{x}}} \right]}\,dx\] is equal to ([x] = greatest integer \[\le \] x)

A)
\[{{\log }_{e}}2\]

B)
\[{{e}^{2}}\]

C)
0

D)
\[\frac{2}{e}\]
View Solution
86)

\[I=\int{\left\{ {{\log }_{e}}{{\log }_{e}}x+\frac{1}{{{({{\log }_{e}}x)}^{2}}} \right\}dx}\] is equal to:

A)
\[x{{\log }_{e}}{{\log }_{e}}x+c\]

B)
\[x{{\log }_{e}}{{\log }_{e}}x-\frac{x}{{{\log }_{e}}x}+c\]

C)
\[x{{\log }_{e}}{{\log }_{e}}x+\frac{x}{{{\log }_{e}}x}+c\]

D)
None of these.
View Solution
87)

\[\int\limits_{0}^{2\pi }{\log \left( \frac{a+b\sec x}{a-b\sec x} \right)}dx=\]

A)
0

B)
\[\pi /2\]

C)
\[\frac{\pi (a+b)}{a-b}\]

D)
\[\frac{\pi }{2}({{a}^{2}}-{{b}^{2}})\]
View Solution
88)

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)
\[K=3\]

B)
\[0\le K<3\]

C)
\[K\le 4\]

D)
\[K=0\]
View Solution
89)

If \[f(x)=a+bx+c{{x}^{2}},\] then what is \[\int_{0}^{1}{f(x)dx}\] equal to?

A)
\[[f(0)+4f(1/2)+f(1)]/6\]

B)
\[[f(0)+4f(1/2)+f(1)]/3\]

C)
\[[f(0)+4f(1/2)+f(1)]\]

D)
\[[f(0)+2f(1/2)+f(1)]/6\]
View Solution
90)

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

B)
2

C)
1/2

D)
0
View Solution

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done Self Evaluation Test - Integrals Total Questions - 90

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Self Evaluation Test - Integrals Total Questions - 90