how did u substitution work in this step?

Question

I was trying to figure out how a double integral was working and i tried putting part of it into an integral calculator, but when they got to this step, they skipped explaining what happened to the $x^2$. I see how they subbed $x^3$ with $u$, but how exactly did the $ x^2$ dissapear? I clicked the steps link but it just showed me how to integrate $x^3$. I'm lost on this part.. please help.

Answer 1

This is what they are doing. They have this substitution $$u = x^3 \\ du=3x^2dx$$ In the integral you have $x^2dx$, you are replacing this with something equal. Which is $$\frac{1}{3}du=x^2dx$$ Now substitute all this in the original integral and you get. $$\frac{1}{3}\int e^udu $$

Answer 2

We have, $$\int x^2e^{x^3}\,dx.$$ Letting, $u=x^3,$ we have, $du=3x^2dx\rightarrow \frac{1}{3}du=x^2dx.$ Note that we already have $x^2dx,$ in our integral such that we can just replace it by $\frac{1}{3}du$, and the integral becomes, $$\int\frac{1}{3}e^u\,du.$$ We can then pull the constant term out of the integral to get, $$\frac{1}{3}\int e^u\,du.$$