# Integral I don't understand?

1. Sep 29, 2011

### QuarkCharmer

1. The problem statement, all variables and given/known data
$$\int \frac{2x}{3x^{2}+10x+3} dx$$
2. Relevant equations

3. The attempt at a solution

I can't think of a U-substitution that would work, nor a trigonometric substitution, or integration by part.

$$\int \frac{2x}{3x^{2}+10x+3} dx$$
$$\int \frac{2x}{(x+3)(3x+1)} dx$$

I factored the denominator out thinking that I could somehow substitute for one product, but that doesn't work clearly. How do you integrate functions like these??

I popped it into wolfram and it had a step about fractional decomposition, but I am having a hard time understanding it and we have not covered it yet in my course.

Here is my go at it:
It has to be in this form right?
$$\frac{2x}{(3+x)(3x+1)} = \frac{A}{3+x} + \frac{B}{3x+1}$$

So now I would multiply the LCD through the equation leaving:
$$2x = A(3x+1) + B(3+x)$$

I don't understand what to do now though?

2. Sep 29, 2011

### George Jones

Staff Emeritus
Multiply out the right side, and then factor out x from all possible terms.

3. Sep 29, 2011

### cepheid

Staff Emeritus
Well, it seems like, in this partial fraction decomposition, you can expand the right hand side of your last equation, collect terms, and then solve for A and B.

4. Sep 29, 2011

### QuarkCharmer

$$2x = A(3x+1) + B(3+x)$$
$$2x = x(3A+B) + A +3B$$

?

5. Sep 29, 2011

### George Jones

Staff Emeritus
left = right.

How many x's on the left? On the right?

What is the constant on the left? On the right?

6. Sep 29, 2011

### QuarkCharmer

So,

The constant is A + 3B

The other equation is 2=(3A+B) ?

I'm guessing I system of equation these guys to find A and B now? What does the constant equal? 0?

7. Sep 29, 2011

### QuarkCharmer

That makes this I believe:

$$\frac{2x}{(3+x)(3x+1)} = \frac{\frac{3}{4}}{3+x} + \frac{\frac{-1}{4}}{3x+1}$$

Does that look correct? I can integrate those.

8. Sep 29, 2011

### cepheid

Staff Emeritus
Yeah, it seems like you've got it. You can always check your answer by doing the reverse (combine the two terms on the right hand side into one fraction with a common denominator and check that the numerator simplifies to 2x).