I need to find know how to integrate $x$ multiplied by a function to a power that is a fraction.

Question

I know how to find integral functions normally, but when I try to find it from say $x\sqrt{4-x^2}$, I get completely lost.This screws me up in both indefinite and definite integration, so please help

Answer 1

We have our integral: $$\int x\sqrt{4-x^2} \ dx$$ We can use u-substitution for this integral. Let: $$u=4-x^2$$ $$du=-2x \ dx$$ Now we can rewrite the integral: $$\int -\dfrac{1}{2}\sqrt{u} \ du$$ $$=-\dfrac{1}{2}\int u^{1/2} \ du$$ $$=-\dfrac{1}{2}\cdot\dfrac{2u^{3/2}}{3}+C$$ $$=-\dfrac{u^{3/2}}{3}+C$$ Reversing the substitution: $$-\dfrac{(4-x^2)^{3/2}}{3}+C$$ $$-\dfrac{\sqrt{(4-x^2)^3}}{3}+C$$ $$\therefore \int x\sqrt{4-x^2}=-\dfrac{\sqrt{(4-x^2)^3}}{3}+C$$