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 ${{\displaystyle \int }}^{\text{}}udv=uv-{{\displaystyle \int }}^{\text{}}vdu$

integration table

a table that lists integration formulas

midpoint rule

a rule that uses a Riemann sum of the form $M_n={\displaystyle \sum }_{i=1}^{n}f(m_i)\text{Δ}x,$ where $m_i$ is the midpoint of the ith subinterval to approximate ${\displaystyle {\int }_a^{b}f\left(x\right)dx}$

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 $\left|\frac{A-B}{A}\right|=\left|\frac{A-B}{A}\right|·100\text{\%}$

Simpson’s rule

a rule that approximates ${\displaystyle {\int }_a^{b}f\left(x\right)dx}$ using the integrals of a piecewise quadratic function. The approximation $S_n$ to ${\displaystyle {\int }_a^{b}f\left(x\right)dx}$ is given by $S_n=\frac{\text{Δ}x}{3}\left(\begin{array}{c}f\left(x_0\right)+4f\left(x_1\right)+2f\left(x_2\right)+4f\left(x_3\right)+2f\left(x_4\right)+4f\left(x_5\right)\\ +\cdots +2f\left(x_{n-2}\right)+4f\left(x_{n-1}\right)+f\left(x_n\right)\end{array}\right)$ trapezoidal rule a rule that approximates ${\displaystyle {\int }_a^{b}f\left(x\right)dx}$ 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