What should I substitute in this Integral?

Question

I have to do homework and need help with this integral. We were supposed to use Fubinis theorem on it, which I think I did. Anyway this is the integral given to us: $$\int_0^1 \int_{x_1}^1 x_2^2 \cdot \sin\left(\frac{2\pi x_1}{x_2}\right)\, dx_2\, dx_1 $$

Which I changed to:$$ \int_0^1 \int_0^{x_2}x_2^2 \cdot \sin\left(\frac{2\pi x_1}{x_2}\right) \,dx_1 \,dx_2$$

Now I've checked this integral on a few calculators and apparently, if it is correct, it's gigantic. So I wanted to ask, if this is correct, what should I substitute to make it actually computable by hand?

Answer 1

Note that

$$\int_{0}^{x_{2}}x_{2}^{2}\sin\left(\dfrac{2\pi x_{1}}{x_{2}}\right){\rm d}x_{1}=-\dfrac{x_{2}^{3}}{2\pi}\cos\left(\dfrac{2\pi x_{1}}{x_{2}}\right)\Bigg|_{x_{1}=0}^{x_{1}=x_{2}}=-\dfrac{x_{2}^{3}}{2\pi}\left[\cos2\pi-\cos0\right]=0$$

Answer 2

Since $x_2$ takes on the value $0$ in this domain, the integral is improper. So you can't switch the order of integration like you have (without proving a Fubini theorem.) You can integrate the original integral by parts twice and then take the limit.

(I'd write more buy my plumber just showed up...)