# Integral x^2(1-x^2)

by Fabio010
Tags: integral, x21x2
 P: 83 People, today i had a exam in math analysis and there was a integral to solve: ∫ x^2√(1-x^2) dx ok, i started to think about the trigonometric substitution. x= sint but, with that substitution now i have a ∫sin^2tcos^2t dt so i have to do something like ∫(1-cos^2t)(cos^2t) dt ok and i thought (no thanks...) i never learned how to solve a integral with the trigonometric formula, so solve something like ∫cos^4t dt takes a lot of time. So i tried t = √(1-x^2) dt/dx = -x/(√(1-x^2) ) So now i have a integral -∫(x^2*t*√(1-x^2))/(x) dt -∫(x^2*t*t)/(x) dt -∫(x*t^2) dt as we know t = √(1-x^2) so x= 1-t^2 -∫((1-t^2)t^2) dt = -∫t^2 - t^4 dt ok now it is easy... Please tell me that i did it in the correct way!
 Sci Advisor HW Helper P: 4,300 Integral x^2(1-x^2) I just made the observation that $$x^2 \sqrt{1 - x^2} \propto x \cdot \frac{d}{dx} (1 - x^2)^{3/2}$$ so maybe you can try partial integration.