Integrating xe^{ax}: A Step-by-Step Solution

[Radarithm]

Homework Statement

Evaluate: $$\int{xe^{ax}}dx$$

Homework Equations

Integration by substitution

The Attempt at a Solution

I'm on a phone at the moment. My work: http://postimg.org/image/v4hdr5uqx/

The correct answer was:
$$\frac{xe^{ax}}{a}-\frac{e^{ax}}{a^2}$$

 

[SteamKing]

You should do this integral by parts. Some of your original substitutions don't look OK.

 

[Radarithm]

http://postimg.org/image/ao3mi4ygz/

I feel like I'm getting closer but I'm still making a dumb mistake. Is it with the derivatives?

 

[eumyang]

If $dv = e^{ax} dx$, then $v = e^{ax}$ is wrong. There are also multiple errors on the second line, but you need to fix what I said first.

 

[Radarithm]

Got it. Thanks for pointing out my mistakes.
$$\int{xe^{ax}}dx$$ $$u=x du=dx dv=e^{ax}dx v=\frac{e^{ax}}{a}$$
$$\frac{xe^{ax}}{a}-\int{\frac{e^{ax}}{a}}dx$$
$$\frac{xe^{ax}}{a}-\int{e^{ax}a^{-1}}dx=\frac{xe^{ax}}{a}-\frac{e^{ax}}{a^2}$$

Power rule + derivative mistake
Thanks for the help.