Question

Evaluate:

`int_0^(1/sqrt(2)) (sin^-1x)/(1 - x^2)^(3/2)*dx`

Answer 1

Let I = `int_0^(1/sqrt(2)) (sin^-1x)/(1 - x^2)^(3/2)*dx`

= `int_0^(1/sqrt(2)) (sin^-1x)/((1 - x^2)sqrt(1 - x^2))*dx`

Put sin^–1 x = t

∴ `(1)/sqrt(1 - x^2)*dx` = dt
Also, x = sin t

When x = `(1)/sqrt(2), t = sin^-1 (1/sqrt(2)) = pi/(4)`

When x = 0, t = sin^–10 = 0

∴ I = `int_0^(pi/4) t/(1 - sin^2t)*dt`

= `int_0^(pi/4) t/(cos^2t)*dt`

= `int_0^(pi/4) t sec^2t*dt`

= `[t int sec^2t*dt]_0^(pi/4) - int_0^(pi/4)[d/dt (t) int sec^2t*dt]*dt`

= `[t tant]_0^(p/4) - int_0^(pi/4) 1*tant*dt`

= `[pi/4 tan  pi/4 - 0] -[log |sect|]_0^(pi/4)`

= `pi/(4) - [log(sec  pi/4) - log (sec 0)]`

= `pi/(4) - [log sqrt(2) - log 1]`

= `pi/(4) - (1)/(2)log2`.                       ...[∵ log 1 = 0]