Integration by Partial Fractions

Q. If $\int \ln (x^2+x)dx=x\ln (x^2+x)+f\left(x\right)+c$, then $f\left(x\right)$ equals:

1. $2x-\ln (x+1)$
2. $-2x-\ln (x+1)$
3. $2x+\ln (x+1)$
4. $-2x+\ln (x+1)$

Q.

Choose a proper fraction out of the following-

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

2. $\frac{3x^3-7x^2+5}{x^2+3}$

3. $\frac{x-3}{x+3}$

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

Q. While using the method of partial fraction, the degree of polynomial in numerator should not be less than that of denominator.

1. True
2. False

Q. If $\int \frac{5\tan x}{\tan x-2}=x+a\ln |b\sin x-d\cos x|+k$, then $a$ is equal to :

1. -1
2. -2
3. 1
4. 2

Q. The number of value(s) of $x$ satisfying the equation $(x+9{)}^2+8|x+9|+7=0$ is

Q. Let $\alpha \in (0,\frac{\pi }{2})$ be fixed. If the integral $\int \frac{\tan x+\tan \alpha }{\tan x-\tan \alpha }dx=A\left(x\right)\cos 2\alpha +B\left(x\right)\sin 2\alpha +C,$ where $C$ is a constant of integration, then the functions $A\left(x\right)$ and $B\left(x\right)$ are respectively:

1. $x-\alpha$ and ${\log }_e|\sin (x-\alpha )|$
2. $x+\alpha$ and ${\log }_e|\sin (x-\alpha )|$
3. $x-\alpha$ and ${\log }_e|\cos (x-\alpha )|$
4. $x+\alpha$ and ${\log }_e|\sin (x+\alpha )|$

Q. Let $f\left(x\right)$ be a quadratic function such that $f\left(0\right)=1$ and $\int \frac{f\left(x\right)}{x^2(x+1{)}^3}dx$ is a rational function. Then the value of $f^{\prime }\left(0\right)$ is

Q. The value of $\int \frac{(x-1)(x-2)(x-3)}{(x-4)(x-5)(x-6)}dx$ is
(where ${}^{\prime }{C}^{\prime }$ is the constant of integration)

1. $x+3\ln |x-4|-24\ln |x-5|+30\ln |x-6|+C$
2. $x+3\ln |x-4|-12\ln |x-5|+30\ln |x-6|+C$
3. $x+3\ln |x-4|-12\ln |x-5|+36\ln |x-6|+C$
4. $x+12\ln |x-4|-24\ln |x-5|+30\ln |x-6|+C$

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

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

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

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

Q.

Let $\int \frac{dx}{x^{2008}+x}=\frac{1}{p}ln\left(\frac{x^q}{1+x^r}\right)+C$ where $p,q,rϵN$ and need not be distinct, then the value of (p+q+r) equals

1. 6024

2. 6022

3. 6021

4. 6020