`inte^(sin^(- 1)x)[1+x/(sqrt(1-x^2))]dx`

Explanation

`"Let I"=int e^(sin^(-1))x[(x+sqrt(1-x^(2)))/(sqrt(1-x^(2)))]dx`
`=int e^(sin^(-1)x)[x+sqrt(1-x^(2))].(1)/(sqrt(1-x^(2)))dx`
`"Put sin"^(-1)x=t " "therefore (1)/(sqrt(1-x^(2)))dx=dt`
`and x=sin t`
`therefore I=int e^(t)[sin t+sqrt(1-sin^(2)t)]dt`
`=int e^(t) [sin t+sqrt(cos^(2)t)]dt`
`=int e^(t) [sin t+cost)]dt`
`"Let f(t)"=sin t" "therefore f'(t)=cos t`
`therefore I=int e^(t)[f(t)+f'(t)]dt`
`=e^(t).f(t)+c=e^(t).sint+c" "=e^(sin^(-1)x).x+c=e^(sin^(-1)x)+c`