Question

Evaluate: `int_0^1 tan^-1(x/sqrt(1 - x^2))dx`.

Answer 1

Let I = `int_0^1 tan^-1(x/sqrt(1 - x^2))dx`

Put x = sin θ    ∴ dx = cos θ dθ

When x = 0, sin θ = 0   ∴ θ = 0  

When x = 1, sin θ = 1   ∴ θ = `π/2`

∴ I = `int_0^(π/2) tan^-1 (sinθ/sqrt(1 - sin^2θ)).cosθ  dθ`

= `int_0^(π/2) tan^-1 (sinθ/cosθ).cosθ  dθ`

= `int_0^(π/2) tan^-1 (tan θ). cosθ  dθ`

= `int_0^(π/2) θ.cosθ  dθ`

= `[θ int cos θ  dθ]_0^(π/2) - int_0^(π/2) [d/(dθ)(θ) int cosθ  dθ]dθ`

= `[θ sin θ]_0^(π/2) - int_0^(π/2) 1.sinθ  dθ`

= `[π/2 sin  π/2 - 0] - int_0^(pi/2) sinθ  dθ`

= `π/2 - [-cosθ]_0^(π/2)`  ...`[∵ sin  π/2 = 1]`

= `π/2 - [-cos  π/2 + cos 0]`

= `π/2 - [-0 + 1]`

= `π/2 - 1`.