`int e^(sin^(-1)x)((log_(e)x)/(sqrt(1-x^(2)))+(1)/(x))dx` is equal to

Options

A. `e^(sin^(-1)xcosx)+C`
B. `-esin^(-1)x.ln(1/x)+C`
C. `(e^(sin^(-1)x))/x+C`
D. `e^(sin^(-1)x)/(sin^(-1)x)+C`

Answer

B

Explanation

`inte^(sin-1)x((lnx)/(sqrt(1-x^(2)))+1/x)dx`
Put `x=sin theta impliesdx=cos theta d theta`
`inte^(theta)((ln sin theta)/(cos theta)+1/(sin theta))cos theta d theta`
`=inte^(theta)(ln sin theta+cot theta)d theta=e^(0).ln(sin theta)+c`
`=e^(sin^(-1)x) lnx+c=-e^(sin^(-1)In(1/x)+C`