in this iterated integral, which variable do i integrate first?

Question

xye^((x^2)y)

how do you do it then? i don't think i can integrate y with what i know, just yet, and with x, it turns out a little funny.

Answer 1

You can do it either way, but one way might be better depending on the limits of your integrals.

If you integrate with repect to x first, you do a substitution u=yx^2,
du=2xy dx to get an anti-derivative of (1/2)e^(x^2 y). The limits of integration on x will determine how hard the next integral (with respect to y) will be.

If you integrate first with respect to y, you have to do an integration by parts with u=xy and dv=e^(x^2 y) dy. Then du=xdy and v=(1/x^2) e^(x^2 y). You will still have to integrate v du to get an anti-derivative of
(y/x)e^(x^2 y) - (1/x^3) e^(x^2 y).
Again, the limits of your integration will dictate how difficult the next integral will be.

If you have to change the order of integration, BE SURE YOU GET THE LIMITS OF INTEGRATION RIGHT.

Answer 2

It doesn't matter. Fubini's theorem assures us that as long as the integral of the absolute value is finite, the result will be the same regardless of which order of integration is used.