Question

The value of the integral $\displaystyle \int e^{3 s i n^{- 1} x}\left(\frac{1}{\sqrt{1 - x^{2}}} + e^{3 c o s^{- 1} x}\right)dx$ is equal to

(where, $c$ is an arbitrary constant)

• A. $\frac{e^{3 \sqrt{s i n^{- 1} x}}}{3}+xe^{\frac{3 \pi }{2}}+c$
• B. $e^{\sqrt{s i n^{- 1} x}}+e^{\pi / 2}+c$
• C. $\frac{e^{3 s i n^{- 1} x}}{3}+xe^{\frac{3 \pi }{2}}+c$
• D. $e^{\frac{\pi }{2}}+e^{x \left(\frac{\pi }{2}\right)}+c$

Answer 1

Answer: (C)

$\displaystyle \int \frac{e^{3 s i n^{- 1} x}}{\sqrt{1 - x^{2}}}dx+$ $\displaystyle \int e^{3 \left(s i n^{- 1} x + c o s^{- 1} x\right)}dx$
$=\displaystyle \int e^{3 t} d t +\displaystyle \int e^{\frac{3 \pi }{2}}dx$ $\left(Put \left(sin\right)^{- 1} x = t\right)$
$=\frac{e^{3 t}}{3}+e^{\frac{3 \pi }{2}}\cdot x+c$