How to use fractional decomposition to integrate rational functions?

[QuarkCharmer]

Homework Statement

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

Homework Equations

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?

 

[George Jones]

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

 

[cepheid]

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.

 

[QuarkCharmer]

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

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

?

 

[George Jones]

left = right.

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

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

 

[QuarkCharmer]

left = right.

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

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

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?

 

[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.

 

[cepheid]

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).