52
MCQ (Single Correct Answer)
If $\int \frac{d x}{(x+2)\left(x^2+1\right)}=a \log \left|1+x^2\right|+b \tan ^{-1} x+\frac{1}{5} \log |x+2|+C$, then
A
$a=\frac{-1}{10}, b=\frac{-2}{5}$
B
$a=\frac{1}{10}, b=-\frac{2}{5}$
C
$a=\frac{-1}{10}, b=\frac{2}{5}$
D
$a=\frac{1}{10}, b=\frac{2}{5}$
53
MCQ (Single Correct Answer)
$\int \frac{x^3}{x+1}$ is equal to
A
$x+\frac{x^2}{2}+\frac{x^3}{3}-\log |1-x|+C$
B
$x+\frac{x^2}{2}-\frac{x^3}{3}-\log |1-x|+C$
C
$x-\frac{x^2}{2}-\frac{x^3}{3}-\log |1+x|+C$
D
$x-\frac{x^2}{2}+\frac{x^3}{3}-\log |1+x|+C$
54
MCQ (Single Correct Answer)
$\int \frac{x+\sin x}{1+\cos x} d x$ is equal to
A
$\log |1+\cos x|+C$
B
$\log |x+\sin x|+C$
C
$x-\tan \frac{x}{2}+C$
D
$x \cdot \tan \frac{x}{2}+C$
55
MCQ (Single Correct Answer)
If $\frac{x^3 d x}{\sqrt{1+x^2}}=a\left(1+x^2\right)^{3 / 2}+b \sqrt{1+x^2}+C$, then
A
$a=\frac{1}{3}, b=1$
B
$a=\frac{-1}{3}, b=1$
C
$a=\frac{-1}{3}, b=-1$
D
$a=\frac{1}{3}, b=-1$
56
MCQ (Single Correct Answer)
$\int_{-\pi / 4}^{\pi / 4} \frac{d x}{1+\cos 2 x}$ is equal to
A
1
B
2
C
3
D
4