Integration by parts problem

• Apr 27th 2011, 05:10 PM
mrantoine
Integration by parts problem
$\int$ (e^(2x))(tan(e^(2x)))^2 dx

I've been trying to integrate this by parts, and I'm failing miserably because no matter how I do it, I just make the integral more complicated or get something that doesn't make any sense at all, so it's pretty useless to put up any attempts at this problem.
• Apr 27th 2011, 05:22 PM
pickslides
Use substitution rather than parts.
• Apr 27th 2011, 05:23 PM
TKHunny
Do you HAVE to use "parts"? That just wouldn't be my first guess. Did you try u = e^(2x)?

One may wish to recall that [tan(x)]^2 + 1 = [sec(x)]^2 and d(tan(x))/dt = [sec(x)]^2
• Apr 27th 2011, 05:43 PM
mrantoine
Alright, I'm pretty rusty on the details of substitution since we learned that several months ago lol but I've got this so far

$\int$ (e^(2x))(tan(e^(2x)))^2 dx
$\int$ (tan u)^2 du

and now im stuck again. If I split tan^2 u into sin^2 u/cos^2 u, then I can't figure out what to substitute next. Sorry I need so much help on this problem, I'm pretty embarrassed considering that (tan u)^2 isn't that elaborate of a function :/

Thanks so much you guys!
• Apr 27th 2011, 05:53 PM
pickslides
The integral of tan^2(x) is tan(x)-x+C, this is a common result.
• Apr 27th 2011, 05:55 PM
mrantoine
Is there a way to arrive at that answer using a method of integration, or is that just something I'll have to memorize before the ap exam?
• Apr 27th 2011, 05:56 PM
pickslides
Quote:

Originally Posted by mrantoine
Is there a way to arrive at that answer using a method of integration, or is that just something I'll have to memorize before the ap exam?

Note that tan^2(x)= sec^2(x)-1

It should hopefully be a little clearer now?!
• Apr 27th 2011, 06:03 PM
mrantoine
ohhh gotcha lol sorry, I'm so tired, I've been studying for 5 hours straight so far today.

So the answer to the whole problem should be tan(e^(2x)) - e^(2x) + c, correct?

p.s. you guys are awesome
• Apr 27th 2011, 07:00 PM
pickslides
Quote:

Originally Posted by mrantoine

So the answer to the whole problem should be tan(e^(2x)) - e^(2x) + c, correct?

This looks good, you can check it yourself by differentiating this.
• Apr 27th 2011, 07:33 PM
TKHunny
Quote:

Originally Posted by mrantoine
Alright, I'm pretty rusty on the details of substitution since we learned that several months ago lol but I've got this so far

$\int$ (e^(2x))(tan(e^(2x)))^2 dx
$\int$ (tan u)^2 du

and now im stuck again. If I split tan^2 u into sin^2 u/cos^2 u, then I can't figure out what to substitute next. Sorry I need so much help on this problem, I'm pretty embarrassed considering that (tan u)^2 isn't that elaborate of a function :/

Thanks so much you guys!

Substitution isn't quite there. You appear to have picked "u" okay. What did you get for "du"? Something fishy going on in there.