Integration by Parts

Q. Find the integral of $ta{n}^{-1}\left(x\right)$

1. $xta{n}^{-1}\left(x\right)-xln|1+x^2|+c$
2. $x^{2}ta{n}^{-1}\left(x\right)-\frac{1}{2}ln|1+x^2|+c$
3. $xta{n}^{-1}\left(x\right)-\frac{1}{2}ln|1+x^2|+c$
4. $xta{n}^{-1}\left(x\right)-\frac{x}{2}ln|1+x^2|+c$

Q.

Why Is Integration Antiderivative?

Q. $\int 2$ cos 2x In (tan⁡x )dx, is equal to

1. sin 2x In (tan x) – 2x + C
2. sin 2xIn (tan x) + 2x + C
3. sin xIn (tan x) – x + C
4. none of these

Q. The integral $\int (1+x-\frac{1}{x})e^{x+\frac{1}{x}}dx$ is equal to:

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

Q. $\int \frac{1-7co{s}^{2}x}{si{n}^{7}xco{s}^{2}x}dx=\frac{f\left(x\right)}{(sinx{)}^7}+C$, then f(x) is equal to

1. sin x
2. cos x
3. tan x
4. cot x

Q. $\int e^{sinx}\left(\frac{xco{s}^{3}x-sinx}{co{s}^{2}x}\right)dx$, is equal to

1. $e^{sinx}(tanx+x)+C$
2. $e^{sinx}(x-secx)+C$
3. $e^{sinx}(secx+tanx)+C$
4. $Noneofthese$

Q. If $\int e^{{\sin }^{2}x}\sin x(\cos x+{\cos }^{3}x)dx=f\left(x\right)+C,$ where $C$ is a constant of integration, then $f\left(x\right)$ equals

1. $\frac{1}{2}{e}^{{\sin }^{2}x}(3-{\sin }^{2}x)$
2. $\frac{1}{2}{e}^{{\sin }^{2}x}(1-\frac{1}{2}{\cos }^{2}x)$
3. $e^{{\sin }^{2}x}(3{\cos }^{2}x+2{\sin }^{2}x)$
4. $e^{{\sin }^{2}x}(2{\cos }^{2}x+3{\sin }^{2}x)$

Q. If $I_n={\int }_0^1\frac{(lnx{)}^n}{\sqrt{x}}dx,nϵN,$ then

1. $I_n$ has finite value for all natural values of '$n$'
2. $\frac{I_2}{I_1}+\frac{I_3}{I_2}+\frac{I_4}{I_3}=-4$
3. $I_4=-9$
4. $I_8=40331$

Q. The value of the integral $\int \sin \left(60x\right){\sin }^{58}xdx$ is
(where ${}^{\prime }{C}^{\prime }$ is the constant of integration)

1. $\frac{\sin \left(59x\right){\sin }^{58}x}{58}+C$
2. $\frac{\sin \left(58x\right){\sin }^{59}x}{60}+C$
3. $\frac{\sin \left(60x\right){\sin }^{59}x}{59}+C$
4. $\frac{\sin \left(59x\right){\sin }^{59}x}{59}+C$

Q. $\int e^{(\sin x-\frac{1}{x})}(1+x\cos x+\frac{1}{x})dx$ is

1. $\cos x.e^{(\sin x-\frac{1}{x})}+c$
2. $\frac{1}{x}.e^{(\sin -\frac{1}{x})}+c$
3. $x.e^{(\sin x-\frac{1}{x})}+c$
4. $e^{(\sin x-\frac{1}{x})}+x+c$

Q. If $\int si{n}^{-1}xco{s}^{-1}xdx=f^{-1}[\frac{\pi }{2}x-x{f}^{-1}(x)-2\sqrt{1-x^2}]+\frac{\pi }{2}\sqrt{1-x^2}+2x+C$ then f(x) is equal to

1. sin 3x
2. sin 2x
3. sin x
4. sin 4x

Q. Let $f,f^{\prime },f^{\prime \prime }$ be continuous in $[0,\ln 2]$ and $f\left(0\right)=0,f^{\prime }\left(0\right)=3,f\left(\ln 2\right)=6,f^{\prime }\left(\ln 2\right)=4$
$\text{and}{\int }_0^{\ln 2}{e}^{-2x}f\left(x\right)dx=3,\text{then}{\int }_0^{\ln 2}{e}^{-2x}{f}^{\prime \prime }\left(x\right)dx$
is

Q. If in the product of two functions, one of the function is not directly integrable$(e.g.\log (x|,{\sin }^{-1}x,\mathrm{etc}.)$ , then we take it as a first function under ILATE.

1. False
2. True

Q.

$I_{m,n}={\int }_0^1{x}^m(\log x{)}^{n}dx$, then $I_{m,n}$ is equal to

1. $\frac{n}{n+1}{I}_{m,n-1}$
   

2. -$\frac{m}{n+1}{I}_{m,n-1}$
   

3. -$\frac{n}{m+1}{I}_{m,n-1}$
   
   

4. $\frac{m}{n+1}{I}_{m,n-1}$

Q. If $S_n=\sum _{r=1}^n{\sin }^{-1}\left(\frac{\sqrt{r}-\sqrt{r-1}}{\sqrt{r(r+1)}}\right)$, then

1. $S_{\infty }=\frac{\pi }{2}$
2. $S_3=\frac{\pi }{6}$
3. $S_{\infty }=\frac{\pi }{4}$
4. $S_3=\frac{\pi }{3}$

Q.

$I_{m,n}={\int }_0^1{x}^m(\log x{)}^{n}dx$, then $I_{m,n}$ is equal to

1. $\frac{n}{n+1}{I}_{m,n-1}$
   

2. -$\frac{m}{n+1}{I}_{m,n-1}$
   

3. -$\frac{n}{m+1}{I}_{m,n-1}$
   
   

4. $\frac{m}{n+1}{I}_{m,n-1}$

Q. If $\int x^5{e}^{-4x^3}dx=\frac{1}{48}{e}^{-4x^3}f\left(x\right)+C$, where $C$ is a constant of integration, then $f\left(x\right)$ is equal to :

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

Q.

$\int \left[f\right(x\left)g^{\prime \prime }\right(x)-f"(x\left)g\right(x\left)\right]dx$ is equal to

1. $\frac{f\left(x\right)}{g^{\prime }\left(x\right)}$

2. $f^{\prime }\left(x\right)g\left(x\right)-f\left(x\right)g^{\prime }\left(x\right)$

3. $f\left(x\right)g^{\prime }\left(x\right)-f^{\prime }\left(x\right)g\left(x\right)$

4. $f\left(x\right)g^{\prime }\left(x\right)+f^{\prime }\left(x\right)g\left(x\right)$

Q.

If $\int g\left(x\right)dx=g\left(x\right)$, then $\int g\left(x\right)\left\{f\right(x)+f^{\prime }(x\left)\right\}dx$ is equal to

1. $g\left(x\right)f\left(x\right)-g\left(x\right)f^{\prime }\left(x\right)+C$

2. $g\left(x\right)f^{\prime }\left(x\right)+C$

3. $g\left(x\right)f\left(x\right)+C$

4. $g\left(x\right)f^2\left(x\right)+C$

Q. Find the integral of the function x sin x

1. $xcosx-sinx$
2. $xcosx+sinx$
3. $-xcosx-sinx$
4. $-xcosx+sinx$

Q. A car moves with a variable acceleration given by $a={\tan }^{-1}\left(t\right)$ where t is the time in seconds. Find the velocity of the car after 10 seconds, if it was initially at rest

1. $10{\tan }^{-1}\left(10\right)-\frac{1}{2}\ln |{10}^2|$
2. $10{\tan }^{-1}(1+{10}^2)-\frac{1}{2}\ln |{10}^2|$
3. ${\tan }^{-1}\left(10\right)-\frac{1}{2}\ln |1+{10}^2|$
4. $10{\tan }^{-1}\left(10\right)-\frac{1}{2}\ln |1+{10}^2|$

Q. If $f\left(x\right)=\{\begin{array}{cc}3(1+|\tan x|{)}^{\frac{\alpha }{|\tan x|}},& -\frac{1}{2}<x<0\\ \beta ,& x=0\\ 3{(1+\left|\frac{\sin x}{3}\right|)}^{\frac{6}{|\sin x|}},& 0<x<\frac{2}{3}\end{array}$ is continuous at $x=0,$ then the ordered pair $(\alpha ,\beta )$ is equal to

1. $(2,\frac{2}{e^2})$
2. $(2,e^2)$
3. $(2,\frac{3}{e^2})$
4. $(2,3e^2)$

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

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

Q.

In some of the cases we can split the integrand into the sum of the two functions such that the integration of one of them by parts produces an integral which cancels the other integral. Suppose we have an integral of the type
$\int \left[f\right(x\left)h\right(x)+g(x\left)\right]dx$
Let $\int f\left(x\right)h\left(x\right)dx=I_1$ and $\int g\left(x\right)dx=I_2$
Integrating $I_1$ by parts, we get
$I_1=f\left(x\right)\int h\left(x\right)dx-\int \left\{f^{\prime }\right(x)\int h(x\left)dx\right\}dx$

$\int \frac{x{e}^x}{(1+x{)}^2}dx$ is equal to

1. $x{e}^x+c$

2. $\frac{e^x}{(x+1{)}^2}+c$

3. $e^x\frac{1}{x+1}+c$

4. $\frac{e^x}{x+1}+c$

Q. If $I_1=\underset{0}{\overset{\pi }{\int }}\frac{\cos x}{(x+2{)}^2}dx,$ $I_2=\underset{0}{\overset{\pi /2}{\int }}\frac{\cos x\sin x}{(x+1)}dx$ and $\lambda I_2=\frac{2}{\mu +\pi }+k-\gamma I_1,$ then

1. $\lambda =4$
2. $\mu =2$
3. $k=1$
4. $\gamma =2$