Integration By Parts: Solving \int e^{2x}sin(e^x)dx

[Sparky_]

Homework Statement

$$\int e^{2x}sin(e^x)dx$$

Homework Equations

Can I make the substitution:

$$w = e^x ; dw = e^x dx$$

Making a "new / simpler" problem:

$$\int w sin(w)$$

The Attempt at a Solution

Using integration by parts on the "new" problem:

$$u = w ; dw = du$$
$$dv = sin (w) ; v = -cos(w)$$

$$\int w sin(w) dw = -w cos(w) +\int cos(w) dw$$

$$= -w cos(w) + sin(w)$$

$$= -e^xcos(e^x) + sin(e^x)$$

is this correct?

This integral is part of a larger problem and this term should "go away" supposedly.

If this is correct (this solution does not simplify to 0), then I will need to post the larger problem -
Thanks for the help
-Sparky

 

[rocomath]

Your problem isn't really that clear to me, is this correct ...

$$\int e^{2x}\sin(e^x)dx$$

 

[Sparky_]

Yes - sorry I was still making my problem presentable when you replied.

It's impressive how quickly you replied.

 

[rocomath]

It's impressive how quickly you replied.

:-]]]

Your initial sub. is perfect so no problem there.

I also get your final answer except with +C at the end which should always be included with indefinite integrals.

 

[rocomath]

Well it checks out and is correct. If you want to know if your Integration is correct, just take the derivative of your answer.

 

[Sparky_]

Sometime soon, I'll try to post the entire problem - it's a differential equation from a book.

I'm not in school but I am trying to brush back up. I have the answer to it - 3 terms summed. I have 4 terms - 3 agree with the 3 - I have an extra.

I'll try to post it perhaps tomorrow. - It's on about 5-6 pages of paper.

I'll condense as appropriate.

thanks for the help.

 

[rocomath]

Gotcha, I'm subscribed.

 

[Sparky_]

rocophysics -
I have posted the "larger" problem in a new thread:
https://www.physicsforums.com/showthread.php?t=207651

Thanks for the help
Sparky -