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This is basically what I did. Not sure if the answer is right or not.

I did the following u substitution.

u = 1 + x^2

u - 1 = x^2

du = 2xdx

x^2du/2 = 2xdu * x^2

x^2du/2 = x^3du

(u - 1)du/2*u^1/2

distributed the u

1/2*u^3/2 - u^1/2

u^2/2 = u

1/2 ∫ u = 1/4*u^2

= 1/4*(1+x^2)^2