Partial Fractions

Here are the steps for computing $\int \frac{P(x)}{Q(x)}\,dx$ , where $P(x)$ and $Q(x)$ are polynomials. First, see if substitution will work. Then check for trig substitution possibilities. If none of these work, or if the trig substitution is too complicated, try partial fraction decomposition.

Partial fraction decomposition process

Check that we have a proper fraction. If not, do long division of polynomials.

Factor $Q(x)$ into a product of linear and irreducible quadratic terms.

Write P(x)Q(x) as a sum of terms with unknown coefficients:
- For every factor of $(x-a)$ in $Q(x)$ , we have a term $\displaystyle\frac{A}{x-a}$ .
- For every repeated linear factor $(x-a)^n$ , we have $\displaystyle\frac{A_1}{(x-a)} + \frac{A_2}{(x-a)^2} + \ldots + \frac{A_n}{(x-a)^n}.$
- For every irreducible quadratic factor $x^2 + bx + c$ , we have $\displaystyle\frac{Ax+B}{x^2+bx+c}.$
- For every repeated irreducible quadratic factor $(x^2+bx+c)^n$ , we have $\displaystyle\frac{A_1 x + B_1}{x^2+bx+c} + \frac{A_2x+B_2}{(x^2+bx+c)^2}+\ldots + \frac{A_nx+B_n}{(x^2+bx+c)^n}.$

Multiply both sides of the equation by $Q(x)$ . Then plug in different values of $x$ to get equations that determine the coefficients. For distinct linear factors, we can always get the coefficients by plugging in the $x$ value that will make the factor zero. For repeated linear factors or for quadratic factors, it can be a little trickier.

If all else fails, compare the coefficients of 1, $x$ , $x^2$ , etc. on both sides to get equations involving $A$ , $B$ , etc.

Integrate. Recall that ∫1x−adx=ln|x−a|+C, and remember substitution (for repeated linear factors), and trig substitution with tangent (irreducible quadratic factors). Integrating other quadratic expressions is done with either a u-substitution or a trig substitution with tangent, after possibly completing the square.

The Fundamental Theorem of Calculus

The Indefinite Integral and the Net Change

Substitution

Area Between Curves

Volumes

Integration by Parts

Integrals of Trig Functions

Trig Substitutions

Partial Fractions

Strategies of Integration

Improper Integrals

Differential Equations

Models of Growth

Infinite Sequences

Infinite Series

Integral Test

Comparison Tests

Convergence of Series with Negative Terms

The Ratio and Root Tests

Strategies for testing Series

Power Series

Representing Functions as Power Series

Taylor and Maclaurin Series

Applications of Taylor Polynomials

Partial Derivatives

Multiple Integrals

Iterated Integrals over Rectangles

Double Integrals over General Regions

A summary

Partial fraction decomposition process