Question

How do I solve this?

Answer 1

a) Recall the integration by parts formula:

$\displaystyle \int u \, dw = uw - \int w \, du$

Then with $u = x$ and $dw = e^x\,dx$ we have

$\displaystyle v = \int e^x x \, dx = xe^x - \int e^x \, dx$

$\implies \boxed{v = xe^x - e^x + C}$

since $w=\int e^x\,dx=e^x$ and $du=dx$. (C is of course an arbitrary constant that can be determined exactly if you know the velocity at some given value of x.)

b) No?

$\displaystyle \int e^xx\,dx = \int xe^x\,dx$

because multiplication is commutative. I get the feeling I'm missing something here...