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 `therefore` x = sin t

`1/(sqrt(1 - x^2)` dx = dt

when x = 0, t = o

x = 1, t = `π/2`

I = `int_0^(π//2)` t² sin t dt 

Integrating by parts, we get

I = `[t^2 (-cos t)_0^(π//2) - int_0^(π//2)` 2t(-cos t) dt

I = `[(-π^2)/4 cos  π/2] - [0] + int_0^(π//2)` 2t cos t dt

I = `int_0^(π//2)` 2t. cos t dt

Integrating by parts, we get

I = `[2t sin]_0^(π//2) - int_0^(π//2)` 2 sin t dt

I = `[2. π/2. sin π/2] - [0] - [-2 cos t]_0^(π//2)`

I = π + 2 `[cos  π/2 - cos 0]`

I = π - 2