Question

Evaluate : `int _0^1 ("x" . ("sin"^-1 "x")^2)/sqrt (1 - "x"^2)` dx

Answer 1

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"`