The value of the integral `intx sin^(-1)xdx` is equal to

Options

A. `(1)/(2)x^(2)sin^(-1)x+(1)/(4)xsqrt(1-x^(2))-(1)/(4)sin^(-1)x+C`
B. `(1)/(2)x^(2)sin^(-1)x-(1)/(4)xsqrt(1-x^(2))-(1)/(4)sin^(-1)x+C`
C. `(1)/(2)x^(2)sin^(-1)x+(1)/(4)xsqrt(1-x^(2))+(1)/(4)sin^(-1)x+C`
D. `(1)/(2)x^(2)sin^(-1)x+(1)/(4)sqrt(1-x^(2))-(1)/(4)sin^(-1)x+C`

Answer

A

Explanation

Let `l=int x sin^(-1)xdx`
On using integration by parts, we get
`l=(sin^(-1)x)(x^(2))/(2)-int(1)/(sqrt(1-x^(2))).(x^(2))/(2)dx`
`rArr" "l=(x^(2))/(2)sin^(-1)x+(1)/(2)int(-x^(2))/(sqrt(1-x^(2)))dx`
`=(x^(2))/(2)sin^(-1)x+(1)/(2)int(1-x^(2)-1)/(sqrt(1-x^(2)))dx`
`rArr" "=(x^(2))/(2)sin^(-1)x+(1)/(2){int(1-x^(2))/(sqrt(1-x^(2)))dx-int(1)/(sqrt(1-x^(2)))dx}`
`rArr ,=(x^(2))/(2)sin^(-1)x+(1)/(2){int sqrt(1-x^(2))dx-int(1)/(sqrt(1-x^(2)))dx}`
`=(x^(2))/(2)sin^(-1)x+(1)/(2)[{(1)/(2)xsqrt(1-x^(2))+(1)/(2)sin^(-1)x}-sin^(-1)x]+C`
`rArr l=(1)/(2)x^(2)sin^(-1)x+(1)/(4)xsqrt(1-x^(2))-(1)/(4)sin^(-1)x+C`