Math 152: Worksheet 9

Method of Partial Fractions

Learning Problems

These problems should be completed on your own. If you need hints on solving a problem, there are some provided with each problem. Click on the word "hint" to view it and again to hide it. They go in increasing order of helpfulness, with the last hint mostly giving away how to do the problem. Try to work from the earlier hints to the later ones, as this will give you the practice you need to succeed in this class.

Problem 1

Evaluate $\displaystyle \int \frac{2x-1}{x^2 - 5x + 6} \ dx $

• Hint 1

  You need to first factor the denominator, which is $(x-3)(x-2)$. What does this mean for the partial fraction decomposition of this rational function?

• Hint 2

  You need to find the coefficients $A$ and $B$ in $2x-1 = A(x-3) + B(x-2)$

• Hint 3

  You should end up with $A = -3$ and $B = 5$

• Hint 4

  The integral you then need to compute is $\displaystyle \int \frac{-3}{x-3} + \frac{5}{x-2}\ dx$

• Answer

  $\displaystyle 5 \ln{|x-3|} - 3 \ln{|x-2|} + C$

Problem 2

Evaluate $\displaystyle \int \frac{x^2 - 8x}{(x+1)(x+4)^3} \ dx $

• Hint 1

  What is the partial fraction decomposition for this function?

• Hint 2

  The decomposition is $\displaystyle \frac{A}{x+1} + \frac{B}{x+4} + \frac{C}{(x+4)^2} + \frac{D}{(x+4)^3}$

• Hint 3

  You can plug in $-1$ and $-4$ to get two of the coefficients, but you'll need to plug in two other numbers to get the last two (try $-2$ and $-3$)

• Hint 4

  The integral you need to compute is $\displaystyle \int \frac{-1/3}{x+4} + \frac{16}{(x+4)^3} + \frac{1/3}{x+1}\ dx$

• Answer

  $\displaystyle -\frac{1}{3} \ln{|x+4|} + \frac{1}{3} \ln{|x+1|} - \frac{1}{8} \frac{1}{(x+4)^2} + C $

Problem 3

Compute $\displaystyle \int \frac{x^2}{(x+1)(x^2 + 1)} dx$

• Hint 1

  The partial fraction decomposition here is of the form $\frac{A}{x+1} + {Bx +C}{x^2 + 1}$

• Hint 2

  You should end up with $A = 1/2$, $B = 1/2$ and $C = -1/2$.

• Hint 3

  The integral to be evaluated is then $\displaystyle \int \frac{1/2}{x+1} + \frac{x/2}{x^2 + 1} - \frac{1/2}{x^2 + 1}\ dx$

• Hint 4

  Your answer should have two logarithm terms and one inverse tangent term.

• Answer

  $\displaystyle \frac{1}{2}\ln{|x+1|} + \frac{1}{4}\ln{|x^2 + 1|} - \frac{1}{2} \tan^{-1}(x) + C$

Problem 4

Compute $\displaystyle \int \frac{3x^2 - 2}{x-4}\ dx$

• Hint 1

  We can't go right into partial fractions here, because the degree in the numerator is higher than the degree in the denominator. How do we fix this?

• Hint 2

  Long division gives that $\displaystyle \frac{3x^2 - 2}{x-4} = 3x + 12 + \frac{46}{x-4}$

• Hint 3

  Then we evaluate the integral; the first part is a polynomial and the second is a logarithm.

• Answer

  $\displaystyle \frac{3}{2}x^2 + 12x + 46\ln{|x-4|} + C$

Problem 5

Compute $\displaystyle \int \frac{10}{(x+1)(x^2 + 9)^2}\ dx $

• Hint 1

  Here we have a repeated quadratic factor, so what does that mean for the partial fraction decomposition of this function?

• Hint 2

  The decomposition you should use is $\displaystyle \frac{A}{x+1} + \frac{Bx + C}{x^2 + 9} + \frac{Dx + E}{(x^2 + 9)^2}$

• Hint 3

  Solving out for the coefficients should give $A = 1/10$, $B = -1/10$, $C = 1/10$, $D = -1$, $E = 1$.

• Hint 4

  You'll need to use trigonometric substitution on the $\frac{1}{(x^2 + 9)^2}$ term, but the rest will be logarithms or inverse tangent.

• Answer

  $\displaystyle \frac{1}{10} \ln{|x+1|} + -\frac{1}{20}\ln{|x^2 + 9|} + \frac{1}{30} \tan^{-1}{\frac{x}{3}} + \frac{1}{2} \frac{1}{x^2 + 9} + \frac{1}{54} \frac{3x}{x^2 + 9} + \frac{1}{54} \tan^{-1}{\frac{x}{3}} + C$

Submission Problems

Problem 1

Compute $\displaystyle \int \frac{3x}{(x^2 + 4)(x-1)}\ dx$.

Problem 2

Compute $\displaystyle \int \frac{x^2 - 3x + 1}{(x-1)^2(x+3)}\ dx$