You can put this solution on YOUR website! To integrate by partial fractions, you need to factor the denominator:
Next we want to rewrite the integrand as a sum of fractions where the denominators of the fractions are the factors of the current denominator. (This is a little more complicated if you have an exponent with a exponent of 2 or more. Fortunately we are not faced with this.) So we want to rewrite as We just have to figure out what values for A and B make this work. To figure this out we need to think, how would we add ? Answer: Get the denominators the same and then add. Now we have to ask: How do we get the denominators the same? Answer: Multiply the numerator and denominator of each fraction by the other fraction's denominator. In our case:
And we want to add up to . This will be true if . (If you prefer formulas instead of all this logic, here is one that will work for a fraction with two factors in the denominator with no factor having an exponent of 2 or more:
(A*(B's denominator) + B*(A's denominator) = (current numerator)
We can solve this equation. Simplify:
Grouping the variable and constant terms and factoring:
Since the right side has no x term, the left side must have one either. This will be true if
A + B = 0
The right side has a constant term of 5 so the left side must have a constant term of 5:
2A + (-3B) = 5
We can now solve this system of A and B. Solving the first equation for B we get:
B = -A
Substituting this into the other equation we get:
2A + (-3(-A)) = 5
2A + 3A = 5
5A = 5
A = 1
B = -A = -(1)
Finally we have our partial fractions!
The new integrals are relatively easy. They are ln's:
Using a properties of logarithms and absolute values, we can combine the the two ln's:
(