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The value of the integral int(0)^(1)xcot...

The value of the integral `int_(0)^(1)xcot^(-1)(1-x^(2)+x^(4))` dx is

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The correct Answer is:
4
`I = int_(0)^(1) x cot^(-1) (1 - x^(2) + x^(4)) dx`
put `x^(2) = t`, 2xdx = dt
`I = (1)/(2) int_(0)^(1) cot^(-1) (1- t + t^(2)) dt, = (1)/(2) int_(0)^(1) tan^(-1) ((1)/(1 + t(t - 1)))dt`
`= (1)/(2) int_(0)^(1) tan^(-1) ((t - (t - 1))/(1 + t (t - 1))) dt, = (1)/(2) int_(0)^(1) {tan^(-1) t + tan^(-1) (1 - t)}dt`
`:. int_(0)^(1) tan^(-1) t dt = int_(0)^(1) tan^(-1) (1 - t) dt, I = int_(0)^(1) tan^(-1) t dt)`
`= t . tan^(-1) t|_(0)^(1) = int_(0)^(1) (t)/(1 + t^(2)) dt = (pi)/(4) - (1)/(2) ln (1 + t^(2)) |_(0)^(1) = (pi)/(4) - (1)/(2) ln 2`
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