# Partial Fractions Homework Solutions

**Hint.** Let u = 2x − 1.

Let u = 2x − 1. Then du = 2 dx, and we get

**Moral. **You can integrate anything that looks like
!

**Hint.** Let u = x^{2} − 5x.

Let u = x^{2} − 5x. Then du = 2x − 5 dx, and we get

**Moral.** Always check to see if you can use
u-substitution before trying anything fancy!

**Hint. **Split up the fraction, then use
u-substitution (with u = x^{2} + 5) on one term and the

following **formula** on the other:

We have

For the first term, let u = x^{2} + 5. Then du = 2x dx, and
the integral becomes

2 ln |u|+C = 2 ln |x^{2}+5|+C. For the second term, use the **formula**
with .
So the answer

to the original integral becomes

**Moral:** You can integrate anything that looks like

**Hint.** Complete the square in the denominator, i.e.
x^{2}+6x+11 = x^{2}+6x+9+2 = (x+3)^{2}+2.

Then let u = x + 3, and apply the technique in problem 3, above.

If u = x + 3 then x = u − 3. du = dx, so completing the square as in the hint,
we have

Now this looks just like the previous problem. Use
u-substitution (or choose a different letter,

since we’re already in u) for the first term, and the inverse-tangent formula
for the second; we get

**Moral:** You can integrate anything that looks like
!

**Hint.
**Perform **polynomial division.
**Check to make sure you get . Integrate.

We verified this long division in class. Now we have

**Moral. **When the integrand is an improper rational
function, perform polynomial division to

rewrite the quotient as a polynomial plus a proper rational function, then apply
the previous

techniques.