Solving the integral $\int r^3e^{-r^2}$ using variable change

Question

We have the following integral:

$\int r^3e^{-r^2} dr$ from $0$ to $\sqrt2$

Normally you would solve this by partial integration, but upon attempting this I get very complex calculations. So instead I was thinking of doing a variable change, but I am not sure what variable change. At first I was thinking changing with $t = r^2$, but I'm not sure how to continue. Any ideas?

Use the substitution you were thinking of, then integrate by parts. — projectilemotion, Aug 17 '17 at 15:08
Motivational! But then I get $\int t^2e^{-t} dr$ and $dt=2rdr$. So $dr = \frac{dt}{2\sqrt t}$. Im not sure how to continue — osk, Aug 17 '17 at 15:09
https://math.stackexchange.com/questions/486006/integrating-int-x3-e-x2dx-by-parts — Hans Lundmark, Aug 17 '17 at 15:11

score 3 · Accepted Answer · answered Aug 17 '17 at 15:10

3

writing $$\int r^2\cdot r e^{-r^2}dr$$ and let $$-r^2=x$$ then we get $$-2rdr=dx$$

answered Aug 17 '17 at 15:10

Dr. Sonnhard Graubner

95,283

Took me a while to process this, but I finally understand! Thank you for your answer! – osk Aug 18 '17 at 18:50

Solving the integral $\int r^3e^{-r^2}$ using variable change

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