Integration by Partial Fractions

Q. 65. Differentiate log sec x using first principle

Q. Let $f\left(x\right)=\left|\begin{array}{ccc}cosx& sinx& cosx\\ cox2x& sin2x& 2cos2x\\ cos3x& sin3x& 3cos3x\end{array}\right|$, then $f^{\prime }\left(\frac{\pi }{2}\right)$ is equal to

1. 6
2. 4
3. 2
4. 8

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. 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. If $\frac{(a^2+1{)}^2}{2a-i}=x+iy,\mathrm{then}{x}^2+y^2$ is equal to
(a) $\frac{(a^2+1{)}^4}{4a^2+1}$
(b) $\frac{(a+1{)}^2}{4a^2+1}$
(c) $\frac{(a^2-1{)}^2}{(4a^2-1{)}^2}$
(d) none of these

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. If $\frac{si{n}^{4}A}{a}+\frac{co{s}^{4}A}{b}=\frac{1}{a+b},$ then $\frac{si{n}^{8}A}{a^3}+\frac{co{s}^{8}A}{b^3}=$

1. 
2. 
3. 
4. 

Q. If (1 + i) (1 + 2i) (1 + 3i) .... (1 + ni) = a + ib, then 2.5.10.17.......(1+n²)=
(a) a − ib
(b) a² − b²
(c) a² + b²
(d) none of these

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. $\frac{1}{15},\frac{1}{3},\frac{4}{5}$

2. $\frac{1}{15},\frac{1}{3},\frac{4}{-5}$

3. $\frac{4}{5},\frac{1}{3},\frac{1}{15}$

4. $\frac{4}{-5},\frac{1}{3},\frac{1}{15}$

Q. $\underset{x\to 0}{\lim }\frac{\sqrt{1+x+x^2}-\sqrt{x+1}}{2x^2}$

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 $e^{f\left(x\right)}=\frac{10+x}{10-x}$, x ∈ (−10, 10) and $f\left(x\right)=kf\left(\frac{200x}{100+x^2}\right)$, then k =
(a) 0.5
(b) 0.6
(c) 0.7
(d) 0.8

Q. 16 Superset and buntons law

Q. Find the derivative of x^5 logx+x^2.3^x)

Q. The number of solutions of the equation $cos\left(\pi \sqrt{x-4}\right)cos\left(\pi \sqrt{x}\right)=1$ is

1. zero
2. 1
3. 2
4. Infinite

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