Question

Evaluate the following : `int (t.sin^-1 t)/sqrt(1 - t^2).dt`

Answer 1

Let I = `int (t.sin^-1 t)/sqrt(1 - t^2).dt`

= `int t.sin^-1 t. 1/sqrt(1 - t^2).dt`

Put sin^–1 t = θ

∴ `1/sqrt(1 - t^2).dt` = dθ
and
t = sin θ
∴ I = `int (sinθ).θdθ`

= `int θ sin θ dθ`

= `θ int sin θ dθ - int [d/(dθ) (θ) int sin θ dθ]dθ`

= `θ (- cos θ) - int 1. (- cosθ)dθ`

= `- θ cosθ + int cosθ  dθ`

= – θ cos θ + sin θ + c

= `- θ.sqrt(1 - sin^2θ) + sin θ + c`

= `- sin^-1 t.sqrt(1 - t^2) + t + c`

= `- sqrt(1 - t^2).sin^-1 t + t + c`.