# Integrate sin(a*x^2) from -a to a

Hey all, this should be a simple integral, but I cant seem to solve it. I tried some simple u-subs and integration by parts, but to no avail. Also, maple throws out some Fresnel function.

$$\int_{-a}^{a} x^{2} cos\left( a x^{2} \right) dx$$

Related Calculus and Beyond Homework Help News on Phys.org
Defennder
Homework Helper
Well, that should tell you it doesn't have an anti-derivative expressible in terms of elementary functions.

I was hoping that perhaps it had a nice representation due to the limits of integration.

edit: I also don't particularly trust maple, as there have been times in the past where it couldn't do integrals that I could. (or perhaps I am using it wrong)

Last edited:
I was hoping that perhaps it had a nice representation due to the limits of integration.
From the looks of it, it's symmetric, so you can change the limits of integration from 0 to a and multiplying your integral by 2. Idk if that helps at all or not :p

This is not a homework question. I am not a student. Please move the question back to the appropriate forum.

Defennder
Homework Helper
Well if it were an odd function, we would be able to conclude that the answer is 0 without having to work out the anti-derivative. But this is an even function.

By the way, if you want your thread to be moved back you should request a PF mentor by private messaging to do so.

>> syms x;
>> f = x^2*cos(x^2);
>> int (f,x)

ans =

1/2*sin(x^2)*x-1/4*2^(1/2)*pi^(1/2)*FresnelS(2^(1/2)/pi^(1/2)*x)

>> int(f,x,-1,1)

ans =

sin(1)-1/2*2^(1/2)*pi^(1/2)*FresnelS(2^(1/2)/pi^(1/2))

Matlab did same ><
I wanted to prove that Matlab is better

"edit: I also don't particularly trust maple, as there have been times in the past where it couldn't do integrals that I could. (or perhaps I am using it wrong)"
sometimes, you need to put the function in integrable form ... (computer's dun think)