Question

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

(where, $c$ is an arbitrary constant)

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

Answer 1

Answer: (C)

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