# Thread: Integration problem help... partial fraction...

1. ## Integration problem help... partial fraction...

I can't seem to figure out this partial fractions problem:

$\displaystyle \int \frac {dx}{3x-3x^2}$

My work so far has involved factoring out a 3x in the denominator so the problem then looks like $\displaystyle \int \frac {1}{3x(1-x)}$, and then multiplying through using the cover up rule, but so far that method hasn't resulted in the answer:

$\displaystyle (\frac {1}{3})\ln \frac {x}{(1-x)} + c$

2. Did you find A and B where $\displaystyle \frac {1}{3x(1-x)} = \frac {A}{3x} +\frac {B}{1-x}$ ?

3. Originally Posted by sgcb
I can't seem to figure out this partial fractions problem:

$\displaystyle \int \frac {dx}{3x-3x^2}$

My work so far has involved factoring out a 3x in the denominator so the problem then looks like $\displaystyle \int \frac {1}{3x(1-x)}$, and then multiplying through using the cover up rule, but so far that method hasn't resulted in the answer:

$\displaystyle (\frac {1}{3})\ln \frac {x}{(1-x)} + c$
I'm not sure what rule your referring to. I would just decompose the function.

$\displaystyle \int\frac{1}{3x(1-x)}dx=\int\left(\frac{A}{3x}+\frac{B}{1-x}\right)dx$

Are you familiar with this method?

$\displaystyle A(1-x)+B(3x)=1$

$\displaystyle x(3B-A)+A=1$

So now you solve the system of equations:
--------------------------

$\displaystyle 3B-A=0$

$\displaystyle A=1$
--------------------------

Substitute those values for A and B into the integral, and it's pretty easy from there.

4. Originally Posted by sgcb
I can't seem to figure out this partial fractions problem:

$\displaystyle \int \frac {dx}{3x-3x^2}$

My work so far has involved factoring out a 3x in the denominator so the problem then looks like $\displaystyle \int \frac {1}{3x(1-x)}$, and then multiplying through using the cover up rule, but so far that method hasn't resulted in the answer:

$\displaystyle (\frac {1}{3})\ln \frac {x}{(1-x)} + c$
note that $\displaystyle \frac{1}{3x(1-x)} = \frac{1}{3} \cdot \frac{1}{x(1-x)}$

and ...

$\displaystyle \frac{1}{x(1-x)} = \frac{1}{x} + \frac{1}{1-x}$

I'm not sure what rule your referring to. I would just decompose the function.
yes both your method and my method output the values A=1 and B=(1/3). But plugging them back into the integral does not yield the correct answer, unless I'm missing a simplification step....

$\displaystyle \ln 3x + (1/3)\ln 1-x + c$

Book:

$\displaystyle (\frac {1}{3})\ln \frac {x}{(1-x)} + c$

6. Recall the fact that

$\displaystyle \ln(a)- \ln(b) = \ln\left(\frac{a}{b}\right)$

7. Originally Posted by pickslides
Recall the fact that

$\displaystyle \ln(a)- \ln(b) = \ln\left(\frac{a}{b}\right)$
I've considered that property, but each time I work out the problem, the natural logs come out as being added and not subtracted.

8. $\displaystyle \frac{1}{x} + \frac{1}{1-x} =\frac{1}{x} + \frac{1}{-(x+1)}= \frac{1}{x} - \frac{1}{x+1}$

now apply the log law.

9. Originally Posted by skeeter
note that $\displaystyle \frac{1}{3x(1-x)} = \frac{1}{3} \cdot \frac{1}{x(1-x)}$

and ...

$\displaystyle \frac{1}{x(1-x)} = \frac{1}{x} + \frac{1}{1-x}$
when the 1/3 is factored out I get:

$\displaystyle 1 = A(1-x) + B(x)$
$\displaystyle = x(B-A) + A$

then
$\displaystyle B - A = 0$
so
$\displaystyle B=A=1$

or am I doing it wrong?

10. Originally Posted by sgcb
I've considered that property, but each time I work out the problem, the natural logs come out as being added and not subtracted.
I suggest skeeter's method as you don't need to solve partial fractions:

$\displaystyle \int \frac {1}{3x(1-x)} = \int \frac{1}{3} + \int \frac{1}{x} + \int \frac{1}{1-x}$

==============================================

I've substituted the values of A and B you found into the expression $\displaystyle \int\left(\frac{1}{3x}+\frac{1}{3-3x}\right)dx$

Note that you integrate that second term you will need to divide by -3 because of the coefficient on the x

$\displaystyle \frac{1}{3}\ln(3x) - \frac{1}{3}\ln(3-3x) + C$ where k is a constant

$\displaystyle \frac{1}{3} \ln \left(\frac{3x}{3(1-x)}\right) +C = \frac{1}{3}\ln \left(\frac{x}{1-x}\right) + C$

11. $\displaystyle \int \frac{1}{x} + \frac{1}{1-x} \, dx = \int \frac{1}{x} - \frac{-1}{1-x} \, dx = \ln|x| - \ln|1-x| + C = \ln\left|\frac{x}{1-x} \right| + C$