Is This Integration by Parts Approach Correct for Solving ∫(xe^x + e^x)dx?

[afcwestwarrior]

Homework Statement

∫e^x+e^x

Homework Equations

∫u dv= uv- ∫v du

The Attempt at a Solution

u= x+e^x
du= e^x

so it would be e^u

integral = e^u

= e^(e^x) +c is that correct, i know the answer is but what i just did

 

[Defennder]

Did you mistype the integral? $$\int 2e^x dx$$?

 

[HallsofIvy]

Homework Statement

∫e^x+e^x

? Surely you don't mean $2\int e^x dx$?

Homework Equations

∫u dv= uv- ∫v du

The Attempt at a Solution

u= x+e^x
du= e^x

Are you now saying the problem is $\int (xe^x+ e^x)dx$?
Then you are not using integration by parts, you are using a simple substitution. Yes, $\int (xe^x+ e^x)dx= \int (x+ e^x)e^x dx$. If you let u= x+ e^x, then du= (1+ e^x) dx, not just e^x dx. And please by sure to include the "dx" in the integral; that may be part of what is confusing you.

so it would be e^u

integral = e^u
= e^(e^x) +c is that correct, i know the answer is but what i just did

Well, you can always check an integration yourself by differentiating.
$$\frac{d}{dx}\left(e^{e^x}\right)= \frac{de^u}{du}\frac{de^x}{dx}$$
with u= e^x
$$= (e^u)(e^x)= (e^{e^x})(e^x)$$
Which is not what you started with.
Did you consider just doing the two integrals separately?
[tex]\int (xe^x+ e^x)dx= \int xe^x dx+ \int e^x dx[/itex]
You should be able to do the second of those directly and the first is a simple integration by parts.