Question

Integrate the following w.r.t.x : `(1)/(x^3 sqrt(x^2 - 1)`

Answer 1

Let I = `int (1)/(x^3 sqrt(x^2 - 1))*dx`
Put x = secθ
∴ dx  secθ tanθ dθ

∴ I = `int (secθ tanθ dθ)/(sec3θ sqrt(sec^2θ - 1)`

= `int (secθ tanθ dθ)/(sec^3θ sqrt(tan^2θ))*dθ`

∴ I = `int cos^2 θ *dθ`

= `(1)/(2) int (1 + cos 2θ)*dθ`

= `(1)/(2) int dθ + 1/2 int cos 2θ*dθ`

= `θ/(2) + (1)/(2)((sin2θ)/2) + c`         ...(1)

∴ x = sec θ
∴ θ = sec^–1x
sin2θ = 2 sinθ cos θ

= `2sqrt(1 - cos^2θ)*cosθ`

= `2sqrt(1 - 1/x^2)(1/x)    ...[because secθ = x  ⇒ cosθ = 1/x]`

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

∴ from (1), we have

I = `(1)/(2)sec^-1 x + 1/2 sqrt(x^2 - 1)/x^2 + c`.