Integration by Substitution

Q. The integral $\underset{1}{\overset{e}{\int }}\{{\left(\frac{x}{e}\right)}^{2x}-{\left(\frac{e}{x}\right)}^x\}{\log }_{e}xdx$ is equal to:

1. $\frac{3}{2}-\frac{1}{e}-\frac{1}{2e^2}$
2. $\frac{3}{2}-e-\frac{1}{2e^2}$
3. $-\frac{1}{2}+\frac{1}{e}-\frac{1}{2e^2}$
4. $\frac{1}{2}-e-\frac{1}{e^2}$

Q.

If $\int {\sin }^{-1}\left(\sqrt{\frac{x}{1+x}}\right)dx=A\left(x\right){\tan }^{-1}\left(\sqrt{x}\right)+B\left(x\right)+C$, where $C$ is a constant of integration, then the ordered pair $\left(A\right(x),B(x\left)\right)$ can be:

1. $\left(x+1,-\sqrt{x}\right)$

2. $\left(x-1,-\sqrt{x}\right)$

3. $\left(x+1,\sqrt{x}\right)$

4. $\left(x-1,\sqrt{x}\right)$

Q.

If $x=\sin t\mathrm{and}y=\sin pt$, then the value of $(1-x^2)\frac{d^{2}y}{d{x}^2}-x\frac{dy}{dx}+p^{2}y=$

1. $0$

2. $1$

3. $-1$

4. $\sqrt{2}$

Q.

The value of $\tan \left[{\tan }^{-1}\left(\frac{1}{2}\right)-{\tan }^{-1}\left(\frac{1}{3}\right)\right]$ is

1. $\frac{5}{6}$

2. $\frac{7}{6}$

3. $\frac{1}{6}$

4. $\frac{1}{7}$

Q. Number of integer(s) for which the function
$f\left(x\right)={\sin }^{-1}\left({\log }_2\left(\frac{x}{3}\right)\right)$ is defined is

Q. The evaluation of $\int \frac{p{x}^{p+2q-1}-q{x}^{q-1}}{x^{2p+2q}+2x^{p+q}+1}dx$ is

1. $-\frac{x^p}{x^{p+q}+1}+C$
   
2. $\frac{x^q}{x^{p+q}+1}+C$
3. $-\frac{x^q}{x^{p+q}+1}+C$
   
4. $\frac{x^p}{x^{p+q}+1}+C$

Q. The value of $\sqrt{2}\int \frac{\sin xdx}{\sin (x-\frac{\pi }{4})}$ is
(where $C$ is constant of integration)

Q. $\int \sin \left({\tan }^{-1}\sqrt{x}\right)dx(x\ge 0)$ is

1. $\sqrt{x^2+x}-\frac{1}{2}\ln \left\{\right(x+\frac{1}{2})+\sqrt{x^2+x}\}+c$
2. $2\sqrt{x^2+x}-\frac{1}{2}\ln \left\{\right(x+\frac{1}{2})+\sqrt{x^2+x}\}+c$
3. $\sqrt{x^2+x}-\ln \left\{\right(x+\frac{1}{2})+\sqrt{x^2+x}\}+c$
4. $\sqrt{x^2+x}+\ln \left\{\right(x+\frac{1}{2})+\sqrt{x^2+x}\}+c$

Q. If the value of $\int \frac{dx}{\cos (3x-2)\cdot \cos (3x+5)}$
is $\frac{f\left(x\right)}{a\sin b}+C,$ then which of the following is/are true
(where $C$ is constant of integration)

Q.

$\int \frac{2sinx}{(3+sin2x)}dx$ is equal to

1. $\frac{1}{2}ln\left|\frac{2+sinx-cosx}{2-sinx+cosx}\right|-\frac{1}{\sqrt{2}}ta{n}^{-1}\left(\frac{sinx+cosx}{\sqrt{2}}\right)+C$
   

2. $\frac{1}{2}ln\left|\frac{2+sinx-cosx}{2-sinx+cosx}\right|-\frac{1}{2\sqrt{2}}ta{n}^{-1}\left(\frac{sinx+cosx}{\sqrt{2}}\right)+C$

3. $\frac{1}{4}ln\left|\frac{2+sinx-cosx}{2-sinx+cosx}\right|-\frac{1}{\sqrt{2}}ta{n}^{-1}\left(\frac{sinx+cosx}{\sqrt{2}}\right)+C$

4. None of these

Q.

If $y={\cos }^{n}x.\sin nx$and $\frac{dy}{dx}=n.{\cos }^{n-1}x\times \cos B$ then $B$ is equals to ?

1. $(n–1)x$

2. $(n+1)x$

3. $nx$

4. $(1–n)x$

Q. If $f\left(x\right)=\sqrt{x},g\left(x\right)=e^{x-1},and\int fog\left(x\right)dx=Afog\left(x\right)+Bta{n}^{-1}\left(fog\right(x\left)\right)+C$, then A + B is equal to

1. 1
2. 2
3. 3
4. \N

Q. $\int \frac{\sqrt{5+x^{10}}}{x^{16}}dx=$

1. $\frac{-1}{75}(1+\frac{5}{x^{10}})+c$
2. $\frac{-1}{50}{(1+\frac{5}{x^{10}})}^{3/2}+c$
3. $\frac{-1}{75}{(1+\frac{5}{x^{10}})}^{3/2}+c$
4. $\frac{-1}{75}{(1+\frac{5}{x^{10}})}^{5/2}+c$

Q. The equation of a curve passing through origin is given by $y=\int x^{3}cos{x}^{4}dx$. If the equation of the curve is written in the form x = g(y), then

1. $g\left(y\right)=\sqrt[3]{3si{n}^{-1}\left(4y\right)}$
2. $g\left(y\right)=\sqrt{si{n}^{-1}\left(4y\right)}$
3. $g\left(y\right)=\sqrt[4]{4si{n}^{-1}\left(4y\right)}$
4. None of these

Q.

$\frac{\cos echx}{\sqrt{(\cos ec{h}^{2}x+1)}}$ equals

1. $\tan hx$

2. $cothx$

3. $sechx$

4. $\cos hx$

Q. How many of the following statements are correct?
$\int tanxdx=ln\left|\right(secx\left)\right|+c\int cotxdx=ln\left|\right(sinx\left)\right|+c\int secxdx=ln\left|\right(tanx\left)\right|+c\int cosec\left(x\right)dx=ln\left|\right(cotx\left)\right|+c$
___

Q. The intergral $\int \frac{se{c}^{2}x}{(secx+tanx{)}^{\frac{9}{2}}}$dx equal (for some arbitrary constant k)

1. $-\frac{1}{(secx+tanx{)}^{\frac{11}{2}}}\{\frac{1}{11}-\frac{1}{7}(secx+tanx{)}^2\}+K$

2. $\frac{1}{(secx+tanx{)}^{\frac{11}{2}}}\{\frac{1}{11}-\frac{1}{7}(secx+tanx{)}^2\}+k$
3. $-\frac{1}{(secx+tanx{)}^{\frac{11}{2}}}\{\frac{1}{11}+\frac{1}{7}(secx+tanx{)}^2\}+k$
4. $\frac{1}{(sexx+tanx{)}^{\frac{11}{2}}}\{\frac{1}{11}+\frac{1}{7}(secx+tanx{)}^2\}+k$

Q. The integral $\underset{\pi /6}{\overset{\pi /3}{\int }}{\sec }^{2/3}x{cosec}^{4/3}xdx$ is equal to :

1. $3^{5/3}-3^{1/3}$
2. $3^{4/3}-3^{1/3}$
3. $3^{5/6}-3^{2/3}$
4. $3^{7/6}-3^{5/6}$

Q. If $f\left(x\right)=\sqrt{x},g\left(x\right)=e^x-1,$ and $\int fog\left(x\right)dx=Afog\left(x\right)+Bta{n}^{-1}\left(fog\right(x\left)\right)+C$, then $A+B$ is equal to

Q. $\int \frac{[x+\sqrt{a^2+x^2}{]}^n}{\sqrt{a^2+x^2}}dx(n\ne 0)=$

1. $\frac{1}{n+1}(x+\sqrt{a^2+x^2}{)}^{n+1}+c$
2. $\frac{1}{n.(x+\sqrt{a^2+x^2}{)}^n}+c$
3. $(x+\sqrt{a^2+x^2}{)}^n+c$
4. $\frac{1}{n}(x+\sqrt{a^2+x^2}{)}^n+c$

Q. If $\int x^5{e}^{-x^2}dx=g\left(x\right)e^{-x^2}+c,$ where $c$ is a constant of integration, then $g(-1)$ is equal to :

1. $-\frac{5}{2}$
2. $-1$
3. $-\frac{1}{2}$
4. $1$

Q. The integral $\int \frac{e^{3{\log }_{e}2x}+5e^{2{\log }_{e}2x}}{e^{4{\log }_{e}x}+5e^{3{\log }_{e}x}-7e^{2{\log }_{e}x}}dx,x>0,$ is equal to :
(where c is a constant of integration)

1. ${\log }_e|x^2+5x-7|+c$
2. $\frac{1}{4}{\log }_e|x^2+5x-7|+c$
3. $4{\log }_e|x^2+5x-7|+c$
4. ${\log }_e\sqrt{x^2+5x-7}+c$

Q. If $\int \frac{\sqrt{x^2+1}}{1-x^2}dx=\frac{1}{\sqrt{2}}f\left(x\right)-g\left(x\right)+C$, where $C$ is a constant of integration, then

1. $f\left(x\right)=\ln \left|\frac{\sqrt{1+x^2}+\sqrt{2}x}{\sqrt{1+x^2}-\sqrt{2}x}\right|$
2. $g\left(x\right)=\ln |x-\sqrt{1+x^2}|$
3. $f\left(x\right)=\ln \left|\frac{\sqrt{1+x^2}-\sqrt{2}x}{\sqrt{1+x^2}+\sqrt{2}x}\right|$
4. $g\left(x\right)=\ln |x+\sqrt{1+x^2}|$

Q. $\int \frac{\sec x(1+\tan x)dx}{(e^{-x}+\sec x)}=f\left(x\right)+c$.
If $f\left(0\right)=\ln \left(2\right)$, then $f\left(\frac{\pi }{4}\right)$ is

1. $\ln (1+\sqrt{2}{e}^{\pi /4})$
2. $\ln \sqrt{2}$
3. $\ln (2\sqrt{2}$)
4. $\ln (\frac{e^{\pi /4}}{\sqrt{2}}+1)$

Q. The value of $\int \frac{e^x+9\cos x-2\sin x+7}{e^x+7\sin x+11\cos x+14}dx$ is
(where $c$ is the constant of integration)

1. $\frac{1}{2}(x+\ln (e^x+7\sin x+11\cos x+14\left)\right)+c$
2. $\frac{1}{2}(x-\ln (e^x+7\sin x+11\cos x+14\left)\right)+c$
3. $x+\frac{1}{2}\ln (e^x+7\sin x+11\cos x+14))+c$
4. $x-\frac{1}{2}\ln (e^x+7\sin x+11\cos x+14))+c$

Q. $\int \frac{dx}{(3+4x^2)\sqrt{(4-3x^2)}}=$

1. $\frac{1}{\sqrt{75}}{\tan }^{-1}\left(\frac{5x}{3\sqrt{4-3x}}\right)+c$
2. $\frac{1}{\sqrt{20}}{\tan }^{-1}\left(\frac{5x}{3\sqrt{4-3x}}\right)+c$
3. $\frac{1}{5\sqrt{3}}{\tan }^{-1}\left(\frac{5x}{\sqrt{12-9x^2}}\right)+c$
4. $\frac{1}{5\sqrt{3}}{\tan }^{-1}\left(\frac{5x}{\sqrt{12-9x^2}}\right)+c$

Q. Evaluate $\int \frac{{\sin }^{-1}\sqrt{x}-{\cos }^{-1}\sqrt{x}}{{\sin }^{-1}\sqrt{x}+{\cos }^{-1}\sqrt{x}}dx$

1. $\frac{1}{\pi }[\sqrt{x-x^2}-(1-2x\left){\cos }^{-1}\sqrt{x}\right]-x+c$
2. $\frac{2}{\pi }[\sqrt{x-x^2}-(1-2x\left){\sin }^{-1}\sqrt{x}\right]-x+c$
3. $\frac{2}{\pi }[\sqrt{x-x^2}-{\sin }^{-1}\sqrt{x}]-x+c$
4. $\frac{2}{\pi }[\sqrt{x-x^2}-(1-2x\left){\sin }^{-1}x\right]-x+c$

Q. The value of the integral $\int \frac{co{s}^{3}x+co{s}^{5}x}{si{n}^{2}x+si{n}^{4}x}$ dx is

1. $sinx-6ta{n}^{-1}\left(sinx\right)+c$
2. $sinx-2(sinx{)}^{-1}+c$
3. $sinx-2(sinx{)}^{-1}-6ta{n}^{-1}(sinx)+c$
4. $sinx-2(sinx{)}^{-1}+5ta{n}^{-1}(sinx)+c$

Q. $\int f\left(x\right)dx=\psi \left(x\right)$, then $\int x^{5}f\left(x^3\right)dx$ is equal to

1. $\frac{1}{3}\left[x^3\psi \right(x^3)-\int x^2\psi \left(x^3\right)dx]+c$
2. $\frac{1}{3}{x}^3\psi \left(x^3\right)-\int x^3\psi \left(x^3\right)dx+c$
3. $\frac{1}{3}{x}^3\psi \left(x^3\right)-\int x^2\psi \left(x^3\right)dx+c$
4. $\frac{1}{3}\left[x^3\psi \right(x^3)-\int x^3\psi \left(x^3\right)dx]+c$

Q. The value of the integral $\int \frac{(x^2-1)dx}{x^3\left(\sqrt{2x^4-2x^2+1}\right)}$ is:

1. $2\sqrt{2-\frac{2}{x^2}+\frac{1}{x^4}}+c$
2. $2\sqrt{2+\frac{2}{x^2}+\frac{1}{x^4}}+c$
3. $\frac{1}{2}\sqrt{2-\frac{2}{x^2}+\frac{1}{x^4}}+c$
4. None of these