Integration by Partial Fractions

Q. The value of the integral $\int \frac{1-x^2}{x(1-2x)}dx$ is
(where $C$ is an arbitrary constant)

1. $\frac{x}{2}+\ln |x|-\frac{3}{4}\ln |1-2x|+C$
2. $\frac{x}{4}+\ln |x|-\frac{4}{3}\ln |1-2x|+C$
3. $\frac{x}{2}+\ln |x|+\frac{4}{3}\ln |1-2x|+C$
4. $\frac{x}{4}-\ln |x|+\frac{3}{4}\ln |1-2x|+C$

Q.

If $\frac{1}{(x-1)(x+2)(2x+3)}$ can be expressed as $\frac{A}{x-1}+\frac{B}{x+2}+\frac{C}{2x+3}$ then what will be the respective values of A, B and C?

1. 

2. 

3. 

4. 

Q.

The value of $\int \frac{x+5}{(x-2{)}^2}dx$ is :

1. $\ln |(x-1)|-\frac{7}{(x-2)}+C$
   

2. $\ln |(x-2)|-\frac{7}{(x-1)}+C$

3. $\ln |(x-2)|-\frac{7}{(x-2)}+C$

4. $\ln |(x-1)|-\frac{7}{(x-1)}+C$

Q.

Choose a proper fraction out of the following-

1. 

2. 

3. 

4. 

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.

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

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. Evaluate $\int \frac{x\mathrm{dx}}{(x-1)(x^2+4)}$

1. $\frac{1}{5}\log \left(x-1\right|+\frac{1}{10}\log (x^2+4)+\frac{2}{5}{\tan }^{-1}\left(\frac{x}{2}\right)+C$
2. $\frac{1}{5}\log \left(x-1\right|-\frac{1}{10}\log (x^2+4)+\frac{2}{5}{\tan }^{-1}\left(\frac{x}{2}\right)+C$
3. $-\frac{1}{5}\log \left(x-1\right|-\frac{1}{10}\log (x^2+4)+\frac{2}{5}{\tan }^{-1}\left(\frac{x}{2}\right)+C$
4. $\frac{1}{5}\log \left(x-1\right|-\frac{1}{10}\log (x^2+4)-\frac{2}{5}{\tan }^{-1}\left(\frac{x}{2}\right)+C$

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

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