### Key Terms

- absolute error
- if $B$ is an estimate of some quantity having an actual value of $A,$ then the absolute error is given by $\left|A-B\right|$

- computer algebra system (CAS)
- technology used to perform many mathematical tasks, including integration

- improper integral
- an integral over an infinite interval or an integral of a function containing an infinite discontinuity on the interval; an improper integral is defined in terms of a limit. The improper integral converges if this limit is a finite real number; otherwise, the improper integral diverges

- integration by parts
- a technique of integration that allows the exchange of one integral for another using the formula ${{\displaystyle \int}}^{\text{}}u\phantom{\rule{0.2em}{0ex}}dv=uv-{{\displaystyle \int}}^{\text{}}v\phantom{\rule{0.2em}{0ex}}du$

- integration table
- a table that lists integration formulas

- midpoint rule
- a rule that uses a Riemann sum of the form ${M}_{n}={\displaystyle \sum}_{i=1}^{n}f({m}_{i})\text{\Delta}x,$ where ${m}_{i}$ is the midpoint of the
*i*th subinterval to approximate ${\int}_{a}^{b}f\left(x\right)dx$

- numerical integration
- the variety of numerical methods used to estimate the value of a definite integral, including the midpoint rule, trapezoidal rule, and Simpson’s rule

- partial fraction decomposition
- a technique used to break down a rational function into the sum of simple rational functions

- power reduction formula
- a rule that allows an integral of a power of a trigonometric function to be exchanged for an integral involving a lower power

- relative error
- error as a percentage of the absolute value, given by $\left|\frac{A-B}{A}\right|=\left|\frac{A-B}{A}\right|\xb7100\text{\%}$

- Simpson’s rule
- a rule that approximates ${\int}_{a}^{b}f\left(x\right)dx$ using the integrals of a piecewise quadratic function. The approximation ${S}_{n}$ to ${\int}_{a}^{b}f\left(x\right)dx$ is given by ${S}_{n}=\frac{\text{\Delta}x}{3}\left(\begin{array}{c}f\left({x}_{0}\right)+4f\left({x}_{1}\right)+2f\left({x}_{2}\right)+4f\left({x}_{3}\right)+2f\left({x}_{4}\right)+4f\left({x}_{5}\right)\\ +\cdots +2f\left({x}_{n-2}\right)+4f\left({x}_{n-1}\right)+f\left({x}_{n}\right)\end{array}\right)$ trapezoidal rule a rule that approximates ${\int}_{a}^{b}f\left(x\right)dx$ using trapezoids

- trigonometric integral
- an integral involving powers and products of trigonometric functions

- trigonometric substitution
- an integration technique that converts an algebraic integral containing expressions of the form $\sqrt{{a}^{2}-{x}^{2}},$ $\sqrt{{a}^{2}+{x}^{2}},$ or $\sqrt{{x}^{2}-{a}^{2}}$ into a trigonometric integral