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Question
Integrate the function in `(x cos^(-1) x)/sqrt(1-x^2)`.
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Solution
Let `I = int (x cos^-1 x)/sqrt(1-x^2) dx`
Put cos-1 x = t
`- 1/sqrt(1-x^2) dx = dt`
`therefore I = - int t cos t dt`
`= - [t int cos t dt - int (d/dt (t)* int cos t dt) dt]`
`= -t sin t + int sin t dt = -t sint - cos t + C`
`= -t sqrt (1 - cos^2 t) - cos t + C`
`= - cos^-1 x sqrt (1 - x^2) - x + C`
`= -[cos^-1 x* sqrt (1 - x^2) + x] + C`
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