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.