The value of `int_(0)^(1)cos^(-1)(x-x^(2))-sqrt((1-x^(2))(2x-x^(2)))dx` is equal to ___________.

Answer

2

Explanation

`I=int_(0)^(1)cos^(-1)((x-x^(2))-sqrt((1-x^(2))(2x-x^(2))))dx`
`=int_(0)^(1)cos^(-1)(x(1-x)-(sqrt(1-x^(2)))(sqrt(1-(1-x)^(2))))dx`
`=int_(0)^(1)(cos^(-1)(x)+cos^(-1)(1-x))dx`
`=2int_(0)^(1)cos^(-1)xdx`
Put `x=cos theta`
`:.I=2int_(0)^(pi//2) theta sin theta d theta`
`=2[-theta cos theta +sin theta ]_(0)^(pi//2)`
`=2`