Solving Integrals with Substitutions: e^x Hint & Attempt at Solution

[jumbogala]

Homework Statement

Integrate

-9e^x - 28 / e^2x + 9e^x + 14

It gives a hint which is substitute u = e^x.

Homework Equations

The Attempt at a Solution

I want to integrate by partial fractions if possible... however before I can do that, I need to make the substitution, and I can't figure out how.

If I take u = e^x, then du=e^x dx .

But I have no e^x dx by itself in my equation to replace?

 

[lanedance]

note you can rewrite it as
$$e^{-x}du = dx$$

so then
$$\frac{du}{u}= dx$$

 

[Mark44]

Here's what I think your integral is, with dx:
$$\int \frac{-9e^x - 28}{e^{2x} + 9e^x +14}dx$$

Since you didn't use any parentheses, it's difficult to tell what the original problem really is, so I wrote the integral as what I thought you meant.

If du = e^x dx, then dx = du/(e^x) = du/u.

Make the substitution, and we'll take it from there.

 

[jumbogala]

Hmm, okay.

So that gives

(-9u-28)/((u^2)+9u+14), that whole thing multiplied by du / u.

Is that right?

 

[lanedance]

sounds alright to me, try and factor the denominator as well

 

[jumbogala]

The denominator factors out into
(u + 2)(u + 7)(u).

So from here I can use partial fractions to integrate, I think.

I will have three terms to integrate, which I'll add together at the end:

A / (u + 2)
B / (u + 7)
C / u

I need to solve for A B and C then integrate. Is this the right apporach, or is there an easier way? Solving for the ABC seems complicated.

 

[Mark44]

Looks good, so far.

 

[jumbogala]

Thanks for your help, both of you. I am not going to continue with the rest of the problem because I know how to solve it, and I still have some other practice questions to do.