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Question
Evaluate : `int _0^1 ("x" . ("sin"^-1 "x")^2)/sqrt (1 - "x"^2)` dx
Solution
Let I = `int _0^1 ("x" . ("sin"^-1 "x")^2)/sqrt (1 - "x"^2)` dx
Put sin-1 x = t
∴ x = sin t
∴ Differentiating w.r.t. x.
`1/sqrt(1 - "x"^2)` dx = dt at x = 0 , t = 0
and at x = 1 , t = `pi/2`
∴ I = `int _0^(pi/2) "sin t" . "t"^2 "dt"`
`= ["t"^2 . int "sin t dt"]_0^(pi/2) - int _0^(pi/2) [("d"/"dx" "t"^2) . int "sin dt"] "dt"`
`= ["t"^2 . (-"cos t")]_0^(pi/2) - int_0^(pi/2) "2t" . (- "cos t") "dt" `
`= [0 - 0] + 2 int _0^(pi/2) "t cos t dt"`
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