Integration by Substitution

Q.

What is the integral of $se{c}^{3}x$?

Q.

${\cos }^{-1}\left(\cos \left(\frac{7\mathrm{\pi }}{6}\right)\right)$

1. $\frac{7\mathrm{\pi }}{6}$

2. $\frac{5\mathrm{\pi }}{6}$

3. $\frac{\mathrm{\pi }}{6}$

4. None of these.

Q.

If $2^x=3^y=6^{-z}$, find the value of $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$.

Q. Let $S\left(x\right)=\int \frac{dx}{e^x+8e^{-x}+4e^{-3x}},R\left(x\right)=\int \frac{dx}{e^{3x}+8e^x+4e^{-x}}$ and $M\left(x\right)=S\left(x\right)-2R\left(x\right).$ If $M\left(x\right)=\frac{1}{2}ta{n}^{-1}\left(f\right(x\left)\right)+c$ then f(0)=

1. $\frac{3}{2}$
2. $\frac{1}{2}$
3. $\frac{5}{2}$
4. $\frac{7}{2}$

Q. $\int x.(x^x{)}^x.(2logx+1)dx$ is equal to

1. $x^{\left(x^x\right)}+C$
2. $(x^x{)}^x+C$
3. $x^x.logx+C$
4. None of these

Q. $\text{If}\int \frac{(3\sin x+4)}{(3+4\sin x{)}^2}=f\left(x\right)+C,\text{where}f\left(0\right)=-\frac{1}{3}\text{and}f\left(\frac{\pi }{2}\right)=0\text{(}C\text{is constant of integration), then}$

1. The minimum value of |$f\left(x\right)$| is $\frac{1}{3}$
2. The minimum value of |$f\left(x\right)$| is 0
3. The value of $f\left(\pi \right)$ is $\frac{1}{3}$
4. The value of $f\left(\pi \right)$ is 0

Q. Evaluate $\int {\sin }^{-1}\left(\frac{2x+2}{\sqrt{4x^2+8x+13}}\right)dx$

1. $(x+1){\tan }^{-1}\left(\frac{2x+2}{3}\right)-\frac{3}{4}\log (4x^2+8x+13)+c$
2. $(x+1){\tan }^{-1}\left(\frac{2}{3}\right)-\frac{3}{4}\log (x^2+8x+13)+c$
3. ${\tan }^{-1}\left(\frac{2x+2}{3}\right)+\frac{3}{4}\log (4x^2+8x+13)+c$
4. none of these

Q. The value of the definite integral $\underset{-1}{\overset{1}{\int }}\frac{dx}{(1+e^x)(1+x^2)}$ is

1. $\frac{\pi }{4}$
2. $\frac{\pi }{8}$
3. $\frac{\pi }{2}$
4. $\frac{\pi }{16}$

Q. $\int \frac{2x+3}{\sqrt{3-x}}dx$ is equal to (where $C$ is integration constant)

1. $18\sqrt{3-x}+\frac{2}{3}{(3-x)}^{\frac{3}{2}}+C$
2. $-18\sqrt{3-x}-\frac{4}{3}{(3-x)}^{\frac{3}{2}}+C$
3. $-18\sqrt{3-x}+\frac{4}{3}{(3-x)}^{\frac{3}{2}}+C$
4. $-18\sqrt{3-x}-\frac{2}{3}{(3-x)}^{\frac{3}{2}}+C$

Q.

Let I $=\int \frac{e^x}{e^{4x}+e^{2x}+1}dx.J=\int \frac{e^{-x}}{e^{-4x}+e^{-2x}+1}dx,$Then, for an arbitrary constant c, the value of J-I equals

1. $\frac{1}{2}log\left(\frac{e^{4x}-e^{2x}+1}{e^{4x}+e^{2x}+1}\right)+c$

2. $\frac{1}{2}log\left(\frac{e^{2x}+e^x+1}{e^{2x}-e^x+1}\right)+c$

3. $\frac{1}{2}log\left(\frac{e^{2x}-e^{2x}+1}{e^{2x}+e^x+1}\right)+c$

4. $\frac{1}{2}log\left(\frac{e^{4x}-e2x+1}{e^{4x}-e^{2x}+1}\right)+c$

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. The value of $\int \frac{dx}{x^n(1+x^n{)}^{\frac{1}{n}}},nϵN$

1. $\frac{1}{1-n}{\{1+\frac{1}{x^n}\}}^{1-\frac{1}{n}}+c$
2. $\frac{1}{1+n}{\{1-\frac{1}{x^n}\}}^{1-\frac{1}{n}}+c$
3. $-\frac{1}{1-n}{\{1-\frac{1}{x^n}\}}^{1-\frac{1}{n}}+c$
4. $\frac{1}{1-n}{\{1+\frac{1}{x^n}\}}^{1-\frac{1}{n}}+c$

Q. If $\int \frac{x^4+1}{x^6+1}dx={\tan }^{-1}\left(f\right(x\left)\right)-\frac{2}{3}{\tan }^{-1}\left(g\right(x\left)\right)+c$, where $c$ is arbitrary constant, then which of the following is/are correct?

1. $f\left(x\right)$ is odd function.
2. $g\left(x\right)$ is odd function.
3. $f\left(x\right)=g\left(x\right)$ has no real roots
4. $\int \frac{f\left(x\right)}{g\left(x\right)}dx=-\frac{1}{x}+\frac{1}{3x^3}+c_1$
   ($c_1$ is a arbitrary constant)

Q. $\int \frac{co{s}^{3}x+co{s}^{5}x}{si{n}^{2}x+si{n}^{4}x}dx$ equals

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

Q. If $\int \frac{x\cos \alpha +1}{(x^2+2x\cos \alpha +1{)}^{3/2}}dx=\frac{f\left(x\right)}{\sqrt{g\left(x\right)+1}}+C$, then $g\left(x\right)-2\cos \alpha f\left(x\right)$ is
(where $C$ is constant of integration)

1. $x^2$
2. $x$
3. $2x^2$
4. $2x$

Q. $\int {\sin }^{-1}\left(\frac{2x+2}{\sqrt{4x^2+8x+13}}\right)dx=f\left(x\right){\tan }^{-1}(g\left(x\right))-\frac{3}{4}\ln (h\left(x\right))+k$
If $h(-1)=9,$ then which of the following option(s) is/are correct?
$($ $k$ is a constant of integration $)$

1. $f\left(x\right)=\frac{2}{3}(x+1)$
2. $g\left(x\right)=\frac{2x+2}{3}$
3. Minimum value of $h\left(x\right)$ is $9$
4. $\frac{d}{dx}h\left(x\right)=8(x+1)$

Q. $\int \frac{dx}{\cos x+\sqrt{3}\sin x}$ equals

1. $\log \tan (\frac{x}{2}+\frac{\pi }{12})+c$
2. $\log \tan (\frac{x}{2}-\frac{\pi }{12})+c$
3. $\frac{1}{2}\log \tan (\frac{x}{2}+\frac{\pi }{12})+c$
4. $\frac{1}{2}\log \tan (\frac{x}{2}-\frac{\pi }{12})+c$

Q. If the integral of the function $\frac{sin\left(lnx\right)}{x}=f\left(x\right)$, then find the value of f(1)
(take constant of integration equal to zero)
___

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

1. $\frac{1}{3}(x+1)$
2. $\frac{2}{3}(x+2)$
3. $\frac{2}{3}(x-4)$
4. $\frac{1}{3}(x+4)$

Q. If $f\left(x\right)=\int \frac{5x^8+7x^6}{{(x^2+1+2x^7)}^2}dx,(x\ge 0)$, and $f\left(0\right)=0$, then the value of $f\left(1\right)$ is :

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

Q. If $\int \frac{8x^{43}+13.x^{38}}{(x^{13}+x^5+1{)}^4}dx=\frac{1}{3}.\frac{x^a}{(x^b+x^c+1{)}^3}+C$ where $a,b,cϵN,(a>b>c$ and where C is a constant of integration), then

1. $a+b=39$
2. $a+b=52$
3. $a+b+c=57$
4. $b-c=8$

Q. $\int \frac{dx}{x^{\frac{1}{2}}(1+x^2{)}^{5/4}}$ is equal to

1. $\frac{-2\sqrt{x}}{\sqrt[4]{4\sqrt{1+x^2}}}+C$
2. $\frac{2\sqrt{x}}{\sqrt[4]{4\sqrt{1+x^2}}}+C$
3. $\frac{-\sqrt{x}}{\sqrt[4]{4\sqrt{1+x^2}}}+C$
4. $\frac{\sqrt{x}}{\sqrt[4]{4\sqrt{1+x^2}}}+C$

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 integral $\int \frac{2x^3-1}{x^4+x}dx$ is equal to : (Here $C$ is a constant of integration)

1. ${\log }_e\left|\frac{x^3+1}{x}\right|+C$
2. $\frac{1}{2}{\log }_e\frac{(x^3+1{)}^2}{\left|x^3\right|}+C$
3. ${\log }_e\frac{|x^3+1|}{x^2}+C$
4. $\frac{1}{2}{\log }_e\frac{|x^3+1|}{x^2}+C$

Q. The integral $\int \frac{{\sec }^{2}x}{(\sec x+\tan x{)}^{9/2}}dx$ equals

1. $-(\sec x-\tan x{)}^{11/2}\{\frac{1}{11}+\frac{1}{7(\sec x-\tan x{)}^2}\}+K$
2. $\frac{1}{(\sec x+\tan x{)}^{11/2}}\{\frac{1}{11}+\frac{1}{7}(\sec x+\tan x{)}^2\}+K$
3. $-\frac{1}{(\sec x+\tan x{)}^{11/2}}\{\frac{1}{11}+\frac{1}{7}(\sec x+\tan x{)}^2\}+K$
4. $(\sec x-\tan x{)}^{11/2}\{\frac{1}{11}+\frac{1}{7(\sec x-\tan x{)}^2}\}+K$

Q. $\int \frac{e^{\sqrt{x}}}{\sqrt{x}}(x+\sqrt{x})dx$ equals

1. $2e^{\sqrt{x}}[x-\sqrt{x}+1]+C$
2. $e^{\sqrt{x}}[x-2\sqrt{x}+1]$
3. $e^{\sqrt{x}}[x+\sqrt{x}]+C$
4. $e^{\sqrt{x}}[x+\sqrt{x}+1]+C$

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. $\int \sin \left({\tan }^{-1}\sqrt{x}\right)dx(x\ge 0)$ 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. 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. The integral $\int \frac{2x^3-1}{x^4+x}dx$ is equal to : (Here $C$ is a constant of integration)

1. ${\log }_e\left|\frac{x^3+1}{x}\right|+C$
2. $\frac{1}{2}{\log }_e\frac{(x^3+1{)}^2}{\left|x^3\right|}+C$
3. ${\log }_e\frac{|x^3+1|}{x^2}+C$
4. $\frac{1}{2}{\log }_e\frac{|x^3+1|}{x^2}+C$