# Integrating 2x/(3x+1)

1. Oct 12, 2013

### lionely

1. The problem statement, all variables and given/known data

Verify, by division, that 2x/(3x+1) = 2/3 - 2/3(3x+1)

Hence, evaluate ∫2x/(3x+1) dx

I don't understand what to, does the question mean to do long division?

Help is much appreciated!

2. Oct 12, 2013

### CompuChip

It's easier to start from the right hand side: take 2/3 - 2/3(3x + 1) and combine the fractions into one. After simplification, you should get 2x/(3x + 1).

3. Oct 12, 2013

### vanhees71

$$\frac{2x}{3x+1}=\frac{2}{3} \cdot \frac{3x}{3x+1}=\frac{2}{3} \cdot \frac{3x+1-1}{3x+1}=\ldots$$

4. Oct 12, 2013

### lionely

Oh thanks, vanhees. But is that still division?

5. Oct 12, 2013

### Ray Vickson

"Division" means re-writing the expression so that in the remainder the numerator has a smaller degree than the denominator. So, if we have f(x) = p(x)/q(x), it is already in "divided" form if deg(p) < deg(q); otherwise (if deg(p) ≥ deg(q)), manipulate the expression to get f(x) = m(x) + r(x)/q(x), with deg(r) < deg(q). (Of course, I mean that p,q,m,r are all polynomials in x.) In your example, deg(p) = 1 = deg(q), so you need to re-write the expression until the remainder has numerator of degree 0 (that is, has the form c/(3x+1) for constant c).

6. Oct 12, 2013

### lionely

Thank you Ray.

7. Oct 12, 2013

### arildno

Or, by polynomial long division

2x:(3x+1)->2/3
2x+2/3
0-2/3

Implies:
2x/(3x+1)=2/3-2/3*(1/(3x+1))

8. Oct 13, 2013

### vanhees71

$$\frac{2x}{3x+1}=\frac{2}{3} \cdot \frac{3x}{3x+1}=\frac{2}{3} \cdot \frac{3x+1-1}{3x+1}=\frac{2}{3} \left (1-\frac{1}{3x+1} \right).$$
Written in this form, the integral is very easy. I don't know, how you call these simple manipulations.

9. Oct 13, 2013

### Ray Vickson

It is impossible to tell which message you are responding to, since you did not use the "Quote" button. Anyway, these manipulations are called simple because they are simple. (However, as far as I can tell, nobody called them simple before you did, so I don't understand your statement!)

10. Oct 14, 2013

### vanhees71

Come on, it is really simple to integrate now!

11. Oct 14, 2013

### CompuChip

Euh, guys... am I the only one who is getting confused here, or are you all replying to the wrong messages. The OT seems to have gotten the hint back in post #6; none of the other posters is the topic starter and knows perfectly well how to solve the exercise, so what is the discussion about?

12. Oct 14, 2013

### arildno

I merely offered an alternative solution procedure!

13. Oct 14, 2013

### D H

Staff Emeritus
That's typical in homework problems. We have a tendency to drag the discussion off topic. The OP hasn't come back since post #6, so presumably the problem has been solved.

That said, I suspect arildno's post #7 is exactly what the instructor intended with the phrase "verify, by division."