Register to reply

Integral x^2(1-x^2)

by Fabio010
Tags: integral, x21x2
Share this thread:
Feb15-12, 08:32 AM
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!
Phys.Org News Partner Science news on
'Smart material' chin strap harvests energy from chewing
King Richard III died painfully on battlefield
Capturing ancient Maya sites from both a rat's and a 'bat's eye view'
Feb15-12, 09:00 AM
Sci Advisor
HW Helper
P: 4,300
You can easily check by differentiating, however I think that you'll find you're off by a factor of x then.

The problem seems to be
as we know t = √(1-x^2) so x= 1-t^2
if t2 = 1 - x2 then x2 = 1 - t2.
I don't know how much that helps you though.
Feb15-12, 09:08 AM
P: 83

what a stupid error.

Damn. ok i should do it by trigonometric substitution.

Feb15-12, 01:20 PM
Sci Advisor
HW Helper
P: 4,300
Integral x^2(1-x^2)

I just made the observation that
[tex]x^2 \sqrt{1 - x^2} \propto x \cdot \frac{d}{dx} (1 - x^2)^{3/2}[/tex]
so maybe you can try partial integration.

Register to reply

Related Discussions
Does path integral and loop integral in a Feynman diagram violate special relativity? General Physics 0
Integral equation with a derivative of the function inside the integral Calculus & Beyond Homework 5
Using polar co-ord. to change double integral into single integral involving only r. Calculus & Beyond Homework 5
Is the ordinary integral a special case of the line integral? Calculus 3
Volume integral to spherical coords to contour integral Calculus & Beyond Homework 4