Partial Fractions: Solving ∫ 5e^2x / (25e^2x - 20e^x +4) dx

[maff is tuff]

Homework Statement

∫ 5e^2x / (25e^2x - 20e^x +4) dx

Homework Equations

The Attempt at a Solution

The attempt at the the solution is in the attachment below. I am stuck at that step. Pretty much can anyone tell me what 'something' should be. I know if the denominator were a reducible quadratic then the numerator would be linear (Ax + B). But what if the denominator is exponential like in my problem?

 

[fzero]

I know if the denominator were a reducible quadratic then the numerator would be linear (Ax + B). But what if the denominator is exponential like in my problem?

It will be something like A e^x +B.

It might be easiest for you to perform a change of variable y=e^x to reduce the problem to a more familiar one.

 

[maff is tuff]

So everywhere I see an e^x I put a y? Then go back in later and put the e^x's back?

 

[fzero]

So everywhere I see an e^x I put a y? Then go back in later and put the e^x's back?

You could do that in order to make the partial fraction computation easier. You could also go further and use

dy = \frac{dy}{dx} dx

to solve for dx in terms of dy, which would allow you to rewrite the integral itself as an integral over the new variable y. You should see this technique come up quite a bit later in your course.

Just to illustrate this with an example, suppose we have

I = \int x^3 dx = \frac{x^4}{4} + c.

If we set y = x^2, we see that dy = 2 x dx, so we can rewrite

I = \int \frac{y ~dy}{2} = \frac{y^2}{4} + c = \frac{x^4}{4} + c.

The reason why the change of variables works is tied to the chain rule for derivatives.

 

[maff is tuff]

Thanks a lot for your help and for taking the time to write out such a detailed explanation. Also, how do you type your math symbols like that? Sorry, I'm relatively new to this site and every time I type math symbols using the ones the forum gives me they don't look like that. Thanks again.

 

[fzero]

Thanks a lot for your help and for taking the time to write out such a detailed explanation. Also, how do you type your math symbols like that? Sorry, I'm relatively new to this site and every time I type math symbols using the ones the forum gives me they don't look like that. Thanks again.

It's called LaTeX code and there's a sticky thread about it here: https://www.physicsforums.com/showthread.php?t=386951 You can also click on the formulas above to see the code that was used to make them.