Question

$ \int e^{x}\left(cosec^{-1}x+\frac{-1}{x\sqrt{x^{2}-1}}\right) \, dx$ is equal to

• A. $ e^x cosec^{-1} x + C $
• B. $ e^x sin^{-1} x + C $
• C. $ e^x sec^{-1} x + C $
• D. $ e^x cos^{-1} x + C $

Answer 1

Answer: (A)

$\int e^{x} \left(cosec^{-1} x+\frac{-1}{x\sqrt{x^{2}-1}}\right) dx$
$=\int e^{x} cosec^{-1}\,x\,dx-\int \frac{e^{x}}{x\sqrt{x^{2}-1}}dx$
$=\left[cosec^{-1}x\cdot e^{x}-\int \frac{-1}{x\sqrt{x^{2}-1}} e^{x}dx\right]$
$-\int \frac{e^{x}}{x\sqrt{x^{2}-1}}dx$
$=e^{x}\cdot cosec^{-1}x+\int \frac{e^{x}}{x\sqrt{x^{2}-1}}dx $
$-\int \frac{e^{x}}{\sqrt{x^{2}-1}}\cdot x\,dx$
$=e^{x}\cdot cosec^{-1}x+c$