What are the steps to finding ∫ x f'(x^2) dx?
I know the answer but I don't know WHY the steps are the following:
= 1/2 ∫ 2x f'(x^2) dx (I'm confused as to where the 1/2 comes from in this step)
= 1/2 ∫ f'(u) du
= 1/2 f(u) + c
= 1/2 f(x^2) + c
Any explanations?
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Verified answer
∫ x f'(x^2) dx
You substitute
u = x^2
du = 2x dx
Solve the second equation for x dx:
du = 2x dx
(1/2) du = x dx
Plug this into the integral:
∫ x f'(x^2) dx
= ∫ (1/2) f'(u) du
= (1/2) ∫ f'(u) du
= (1/2) f(u) +c
= (1/2) f(x^2) + c
I hope that helps!