On an indefinite integral

Question

I just had my midterm exam in multivariable calculus and this was one of the final questions, i kinda struggled with. But I'm supposed to find the indefinite integral of:

$$\int\int_{R^{2}}(x^2+y^2)e^{-(x^2+y^2)^2}$$

I tried to transform it into a polar coord integral, but I didn't know how to continue, could some one show me or tell me what this would equal to? $$\int^{2\pi}_{0}\int^1_0r^3e^{-r^4}drd\theta$$

Answer 1

$$=\left(\int_0^{2\pi} d\theta\right)\left(\frac {-1}{4}\int _0^1 -4r^3e^{-r^4}dr\right) $$ Let $-r^4=u$ Hence $-4r^3dr=du$ $$=\frac {-\pi}{2} \int_0^{-1} e^u du$$ $$=\frac {-\pi(e-1)}{2e}$$

Answer 2

Hint: $(e^{-r^4})'=-4r^3e^{-r^4}$ apply Fubini, by integrating relatively to $r$ first.

Answer 3

HINT

Note that

$$\int\int_{R^{2}}(x^2+y^2)e^{-(x^2+y^2)^2} =\int^{2\pi}_{0}\int^{\infty}_0r^3e^{-r^4}drd\theta=2\pi\int^{\infty}_0r^3e^{-r^4}dr $$

and

$$(e^{-r^4})'=-4r^3e^{-r^4}$$

Answer 4

Hint: by substitution: $$t=r^4; dt=4r^3dr$$