49
MCQ (Single Correct Answer)
$\frac{d x}{\sin (x-a) \sin (x-b)}$ is equal to
A
$\sin (b-a) \log \left|\frac{\sin (x-b)}{\sin (x-a)}\right|+C$
B
$\operatorname{cosec}(b-a) \log \left|\frac{\sin (x-a)}{\sin (x-b)}\right|+C$
C
$\operatorname{cosec}(b-a) \log \left|\frac{\sin (x-b)}{\sin (x-a)}\right|+C$
D
$\sin (b-a) \log \left|\frac{\sin (x-a)}{\sin (x-b)}\right|+C$
50
MCQ (Single Correct Answer)
$\int \tan ^{-1} \sqrt{x} d x$ is equal to
A
$(x+1) \tan ^{-1} \sqrt{x}-\sqrt{x}+C$
B
$x \tan ^{-1} \sqrt{x}-\sqrt{x}+C$
C
$\sqrt{x}-x \tan ^{-1} \sqrt{x}+C$
D
$\sqrt{x}-(x+1) \tan ^{-1} \sqrt{x}+C$
51
MCQ (Single Correct Answer)
$\int \frac{x^9}{\left(4 x^2+1\right)^6} d x$ is equal to
A
$\frac{1}{5 x}\left(4+\frac{1}{x^2}\right)^{-5}+C$
B
$\frac{1}{5}\left(4+\frac{1}{x^2}\right)^{-5}+C$
C
$\frac{1}{10 x}(1+4)^{-5}+C$
D
$\frac{1}{10}\left(\frac{1}{x^2}+4\right)^{-5}+C$
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
$x-\frac{x^2}{2}+\frac{x^3}{3}-\log |1+x|+C$