Homework Help: Some integral questions

I think it would be easier to do the first one with integration by parts first
[tex]\int \frac {e^{2x}} {1 + e^x} dx = \int e^x * \frac {e^x} {1 + e^x} dx[/tex]

Try something similar for the second one where you can integrate the part on the right with a substitution.

 

...but if you wanted to use just straight substitution, then what you guessed u=1+e^x will work. Bohrok's splitting of e^2x into two factors of e^x is part of the solution. Don't forget that you have to replace dx with (something)*du. That should take care of one of your upstairs e^x. The third step is to write the left over e^x in terms of the u substitution you have picked. What you get in terms of u should be pretty easy to integrate.

 

Look at the square root term. That should remind you of a special kind of substitution technique. Something Trig'ish perhaps?

 

"x^3/sqrt(1-x^2) using integration by parts"


The integral of x/sqrt(1-x^2) is proportional to sqrt(1-x^2), so it makes sense to do partial integration by integrating this factor. Then you need to know the integral of sqrt(1-x^2) times the derivative of x^2, i.e. 2xsqrt(1-x^2) which is, of course, proportional to
(1-x^2)^(3/2).

 

Yep, that works too. However, doing integration by parts on the trig function that you get seems less...erm...cryptic to me, than trying to guess dv=x*dx / sqrt() and integrating that. Unless you happen to know what that integral is, you still need to do some substitution technique to get that answer. In either case, substitution then parts, or parts then substitution gets the same answer. Ain't integration fun :)