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...