Strategies of Integration

Partial fractions is a way to integrate functions of the form $\displaystyle\frac{P(x)}{Q(x)}$ , where $P(x)$ and $Q(x)$ are polynomials (also called rational functions).

The key is to decompose $\displaystyle\frac{P(x)}{Q(x)}$ into a sum of simpler fractions, which are related to the terms in the factorization of $Q(x)$ .

Example:

$\frac{2}{x^2-1} = \frac{1}{x-1} - \frac{1}{x+1}$

What to look for

If the degree of $P(x)$ is greater or equal to the degree of $Q(x)$ , then we need to use long division.

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 quadratic factor $x^2 + bx + c$ , we have $\displaystyle\frac{Ax+B}{x^2+bx+c}.$

For every repeated quadratic factor $(x^2+bx+c)^n$ , we have $\displaystyle\frac{A_1 x + B_1}{x^2+bx+c} + \frac{A_2x+B_2}{(x^2+bx+c)^2}+\ldots + \frac{A_nx+B_n}{(x^2+bx+c)^n}.$

Finally, we have to integrate the resulting terms. Linear factors give logs. Quadratic factors give a combination of logs and arctangents, thanks to what you learned about trig substitutions.

Example:

$\int \frac{x^3}{x^2- 1}\,dx = \int \left(x + \frac{x}{x^2- 1}\right)\,dx= \int x\,dx+ \frac{1}{2}\int\left( \frac{1}{x-1} + \frac{1}{x+1} \right)\,dx$

$\qquad\qquad\qquad= \frac{x^2}{2}+ \frac{1}{2}\Bigl(\ln\lvert x-1\rvert +\ln\lvert x+1\rvert \Bigr)+ C$

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

Partial Fractions

What to look for