a) Recall the integration by parts formula:
[tex]\displaystyle \int u \, dw = uw - \int w \, du[/tex]
Then with [tex]u = x[/tex] and [tex]dw = e^x\,dx[/tex] we have
[tex]\displaystyle v = \int e^x x \, dx = xe^x - \int e^x \, dx[/tex]
[tex]\implies \boxed{v = xe^x - e^x + C}[/tex]
since [tex]w=\int e^x\,dx=e^x[/tex] and [tex]du=dx[/tex]. (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?
[tex]\displaystyle \int e^xx\,dx = \int xe^x\,dx[/tex]
because multiplication is commutative. I get the feeling I'm missing something here...
