Finding the Antiderivative using Substitution Rule.

Question

Answer 1

Let u = 1+x². So, du = 2xdx. Therefore, xdx = (1/2)du.

∫x√(1+x²)dx = ∫ u^1/2(1/2)du = (1/2)(2/3)u^3/2 + C = (1/3)(1 + x²)^3/2 + C

Answer 2

let u = 1+x^2 then du = 2x

integral of u^1/2 du = u^3/2/3/2 + a constant

= (2/3)u^3/2 + a constant

= (2/3)(1+ x^2)^3/2 + a constant