Integration by parts

Integrate using integration by parts:

$F(y) = (y+1)e^{-y}$

Find:

Evaluate the $\int_{a=0}^{b=\infty}F(y)\;dy$ using integration by parts.

Thus far, I've distributed the $e^y$ term and split this into two integrals. One of these integrals becomes trivially easy to solve. The Other integral, the integral of $ye^{-y}$, I solved using integration by parts. I think. However, it's possible I'm making some mistakes somewhere.

$\left[-ye^{-y} - 2e^{-y}\right]_{a=0}^{b=\infty}$

My concern is that when evaluating with the infinity term I encounter an indeterminate form, do I not? Also, I'm curious if I can say that $e^{-y}$ where $y=\infty$ is defined at all. Wouldn't we only be able to talk about what happens in the limit?

• Distributing the exponential is unnecessary and just making extra work. Let $u=x+1,dv=e^{-x}dx,du=dx,v=-e^{-x}$. – bof Aug 27 '14 at 22:27
$\int_0^{\infty}(x+1)e^{-x}dx=-(x+1)e^{-x}\Big|_0^{\infty}+\int_0^{\infty}e^{-x}dx=1+1=2$
Your answer is right. As for evaluating the integral, the limit at $\infty$ does exist, because $e^{-y}$ converges to zero as $y \rightarrow \infty.$ I think you should get an answer of $2$ after evaluating the integral, but you should check!