Section 7.3 - Partial Fractions
Instructions
- The first video below explains the concepts in this section.
- This page also includes exercises that you should attempt to solve yourself. You can check your answers and watch the videos explaining how to solve the exercises.
Concepts
- Integration Rational Functions by using the method of partial fractions to write the rational function as a sum of simpler rational functions
- Using long division of polynomials to simplify a rational function before using partial fractions
- Determining the correct partial fraction decomposition to use for a rational function
Exercises
Directions: You should try to solve each problem first, and then click "Reveal Answer" to check your answer. You can click "Watch Video" if you need help with a problem.
1. Find \(\displaystyle{\int \displaystyle{{4x+1}\over{2x^3+5x^2+3x}}\,dx}\)
\( \displaystyle \frac{1}{3}\ln |x| -\frac{10}{3} \ln |2x+3| + 3\ln |x+1| + C\)
2. Find \(\displaystyle{\int_{-1}^0 \displaystyle{{x^3}\over{x+2}}\,dx}\)
\( \displaystyle -8\ln 2 + \frac{16}{3}\)
3. Find \(\displaystyle{\int \displaystyle{{1}\over{(x-1)^2(x+4)}}\,dx}\)
\( \displaystyle -\frac{1}{25}\ln|x-1|-\frac{1}{5(x-1)}+\frac{1}{25}\ln|x+4|+C\)
4. Find \(\displaystyle{\int \displaystyle{{3x^3+18}\over{x^2(x^2+9)}}\,dx}\)
\( \displaystyle -\frac{2}{x}+\frac{3}{2}\ln\left| x^2+9\right| -\frac{2}{3}\arctan\left(\frac{x}{3}\right) + C\)
5. Find \(\displaystyle{\int_0^1 \displaystyle{{x+6}\over{(x^2+1)(x^2+4)}}\,dx}\)
\( \displaystyle \frac{1}{6}\ln 2 +\frac{\pi}{2}-\frac{1}{6}\ln 5 - \arctan \frac{1}{2} + \frac{1}{6}\ln 4\)