Please solve it step by step. Integrate:

Question

∫(x^3/(√(x^2+1))dx

Answer 1

x = tanθ
x^2+1 = sec^2θ
dx = sec^2θ dθ

and now your integrand becomes

tan^3θ/secθ * sec^2θ dθ
= tan^3θ secθ dθ
= tan^2θ (secθ tanθ dθ)
= (sec^2θ - 1) (secθ tanθ dθ)

let u = secθ
du = secθ tanθ dθ
and now you have just

(u^2-1) du