Skip to main content
\(\require{mathrsfs}\require{cancel}\newcommand{\dee}[1]{\mathrm{d}#1} \newcommand{\half}{ \frac{1}{2} } \newcommand{\ds}{\displaystyle} \newcommand{\ts}{\textstyle} \newcommand{\es}{ {\varnothing}} \newcommand{\st}{ {\mbox{ s.t. }} } \newcommand{\pow}[1]{ \mathcal{P}\left(#1\right) } \newcommand{\set}[1]{ \left\{#1\right\} } \newcommand{\lin}{{\text{LIN}}} \newcommand{\quot}{{\text{QR}}} \newcommand{\simp}{{\text{SMP}}} \newcommand{\diff}[2]{ \frac{\mathrm{d}#1}{\mathrm{d}#2}} \newcommand{\bdiff}[2]{ \frac{\mathrm{d}}{\mathrm{d}#2} \left( #1 \right)} \newcommand{\ddiff}[3]{ \frac{\mathrm{d}^#1#2}{\mathrm{d}{#3}^#1}} \renewcommand{\neg}{ {\sim} } \newcommand{\limp}{ {\;\Rightarrow\;} } \newcommand{\nimp}{ {\;\not\Rightarrow\;} } \newcommand{\liff}{ {\;\Leftrightarrow\;} } \newcommand{\niff}{ {\;\not\Leftrightarrow\;} } \newcommand{\De}{\Delta} \newcommand{\bbbn}{\mathbb{N}} \newcommand{\bbbr}{\mathbb{R}} \newcommand{\bbbp}{\mathbb{P}} \newcommand{\cI}{\mathcal{I}} \newcommand{\cR}{\mathcal{R}} \newcommand{\cV}{\mathcal{V}} \newcommand{\Si}{\Sigma} \newcommand{\arccsc}{\mathop{\mathrm{arccsc}}} \newcommand{\arcsec}{\mathop{\mathrm{arcsec}}} \newcommand{\arccot}{\mathop{\mathrm{arccot}}} \newcommand{\erf}{\mathop{\mathrm{erf}}} \newcommand{\smsum}{\mathop{{\ts \sum}}} \newcommand{\atp}[2]{ \genfrac{}{}{0in}{}{#1}{#2} } \newcommand{\ave}{\mathrm{ave}} \newcommand{\llt}{\left \lt } \newcommand{\rgt}{\right \gt } \newcommand{\YEaxis}[2]{\draw[help lines] (-#1,0)--(#1,0) node[right]{$x$};\draw[help lines] (0,-#2)--(0,#2) node[above]{$y$};} \newcommand{\YEaaxis}[4]{\draw[help lines] (-#1,0)--(#2,0) node[right]{$x$};\draw[help lines] (0,-#3)--(0,#4) node[above]{$y$};} \newcommand{\YEtaxis}[4]{\draw[help lines] (-#1,0)--(#2,0) node[right]{$t$};\draw[help lines] (0,-#3)--(0,#4) node[above]{$y$};} \newcommand{\YEtaaxis}[4]{\draw[help lines, <->] (-#1,0)--(#2,0) node[right]{$t$}; \draw[help lines, <->] (0,-#3)--(0,#4) node[above]{$y$};} \newcommand{\YExcoord}[2]{\draw (#1,.2)--(#1,-.2) node[below]{$#2$};} \newcommand{\YEycoord}[2]{\draw (.2,#1)--(-.2,#1) node[left]{$#2$};} \newcommand{\YEnxcoord}[2]{\draw (#1,-.2)--(#1,.2) node[above]{$#2$};} \newcommand{\YEnycoord}[2]{\draw (-.2,#1)--(.2,#1) node[right]{$#2$};} \newcommand{\YEstickfig}[3]{ \draw (#1,#2) arc(-90:270:2mm); \draw (#1,#2)--(#1,#2-.5) (#1-.25,#2-.75)--(#1,#2-.5)--(#1+.25,#2-.75) (#1-.2,#2-.2)--(#1+.2,#2-.2);} \newcommand{\IBP}[7]{ \begin{array}{|c | l | l |} \hline \color{red}{\text{Option 1:}} & u=#2 &\color{red}{\dee{u}=#3 ~ \dee{#1}} \\ & \dee{v}=#5~\dee{#1} &\color{red}{v=#7} \\ \hline \color{blue}{\text{Option 2:}} & u=#5 &\color{blue}{\dee{u}=#6 ~ \dee{#1}} \\ &\dee{v}=#2 \dee{#1} &\color{blue}{v=#4} \\ \hline \end{array} } \renewcommand{\textcolor}[2]{{\color{#1}{#2}}} \newcommand{\trigtri}[4]{ \begin{tikzpicture} \draw (-.5,0)--(2,0)--(2,1.5)--cycle; \draw (1.8,0) |- (2,.2); \draw[double] (0,0) arc(0:30:.5cm); \draw (0,.2) node[right]{$#1$}; \draw (1,-.5) node{$#2$}; \draw (2,.75) node[right]{$#3$}; \draw (.6,1.1) node[rotate=30]{$#4$}; \end{tikzpicture}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \)

Section1.8Trigonometric Integrals

Integrals of polynomials of the trigonometric functions \(\sin x\text{,}\) \(\cos x\text{,}\) \(\tan x\) and so on, are generally evaluated by using a combination of simple substitutions and trigonometric identities. There are of course a very large number  1 The more pedantic reader could construct an infinite list of them. of trigonometric identities, but usually we use only a handful of them. The most important three are:

Notice that the last two lines of Equation 1.8.3 follow from the first line by replacing either \(\sin^2x\) or \(\cos^2x\) using Equation 1.8.1. It is also useful to rewrite these last two lines:

These last two are particularly useful since they allow us to rewrite higher powers of sine and cosine in terms of lower powers. For example:

\begin{align*} \sin^4(x) &= \left[ \frac{1-\cos(2x)}{2} \right]^2 &\text{by Equation }\knowl{./knowl/eq_TRGINTtrigidentityF.html}{\text{1.8.4}}\\ &= \frac{1}{4} - \frac{1}{2} \cos(2x) + \frac{1}{4}\underbrace{\cos^2(2x)}_{\text{do it again}} & \text{use Equation }\knowl{./knowl/eq_TRGINTtrigidentityG.html}{\text{1.8.5}}\\ &= \frac{1}{4} - \frac{1}{2} \cos(2x) + \frac{1}{8}\left(1 + \cos(4x) \right)\\ &= \frac{3}{8} - \frac{1}{2} \cos(2x) + \frac{1}{8}\cos(4x) \end{align*}

So while it was hard to integrate \(\sin^4(x)\) directly, the final expression is quite straightforward (with a little substitution rule).

There are many such tricks for integrating powers of trigonometric functions. Here we concentrate on two families

\begin{align*} \int \sin^mx \cos^nx \dee{x} &&\text{and}&& \int \tan^mx \sec^nx \dee{x} \end{align*}

for integer \(n,m\text{.}\) The details of the technique depend on the parity of \(n\) and \(m\) — that is, whether \(n\) and \(m\) are even or odd numbers.