People, today i had a exam in math analysis and there was a integral to solve:(adsbygoogle = window.adsbygoogle || []).push({});

∫ 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!

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

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