Key Techniques for Integration: Substitution and Parts - Integral Practice

[Emethyst]

Homework Statement

Find the indefinate integral of e^2x/(1+e^x) using integration by substitution, and find the indefinate integral of x^3/sqrt(1-x^2) using integration by parts

Homework Equations

Integration by substitution, integration by parts

The Attempt at a Solution

The problems I'm having on these questions is choosing what to use for variables. In the first question, I decided to use u=1+e^x, but am not sure what to do about the e^2x. I know I will have to square something but am not sure what. For question 2, I have no idea what to use. I have tried making u=x^2, dv=dx and several other options but none of them seem to work. If anyone can guide me along to making the correct variable choices for these questions it would be greatly appreciated, thanks in advance.

 

[Bohrok]

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

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

 

[Chrisas]

...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.

 

[Emethyst]

Ahh I see that now, thank you, though how about the second question, what can I do for that one?

 

[Chrisas]

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

 

[Count Iblis]

"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).

 

[Chrisas]

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 :)