Integrate expressions containing products and powers of sine and cosine
Integrate expressions containing products and powers of tangent and secant
Use reduction formulas to simplify and solve trigonometric integrals
Integrate expressions containing square roots of sums or differences of squares
Integrating Products and Powers of [latex]\sin{x}[/latex] and [latex]\cos{x}[/latex]
In this section, you’ll learn how to integrate products of trigonometric functions—skills that will be essential for advanced techniques like trigonometric substitution and coordinate systems you’ll encounter later. We’ll start with products of [latex]\sin{x}[/latex] and [latex]\cos{x}[/latex] and build from there.
A key idea behind the strategy used to integrate combinations of products and powers of [latex]\sin{x}[/latex] and [latex]\cos{x}[/latex] involves rewriting these expressions as sums and differences of integrals of the form [latex]\displaystyle\int\sin^{j}x\cos{x}dx[/latex] or [latex]{\displaystyle\int}{\cos}^{j}x\sin{x}dx[/latex]. After rewriting these integrals, we evaluate them using u-substitution.
Before describing the general process in detail, let’s take a look at a couple of examples.
Watch the following video to see the worked solution to the above example.
For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.
In addition to the technique of [latex]u-[/latex] substitution, the problems in this section and the next make frequent use of the Pythagorean Identity and its implications for how to rewrite trigonometric functions in terms of other trigonometric functions. Here’s a quick review of these essential relationships.
For any angle [latex]x[/latex]:
[latex]\sin^2 x + \cos^2 x = 1[/latex]
Subtracting by [latex]\sin^2 x[/latex] allows a square power of cosine in terms of sine:
[latex]\cos^2 x = 1-\sin^2 x[/latex]
Subtracting instead by [latex]\cos^2 x[/latex] allows a square power of sine to be written in terms of cosine:
To convert this integral to integrals of the form [latex]{\displaystyle\int}{\cos}^{j}x\sin{x}dx[/latex], rewrite [latex]{\sin}^{3}x={\sin}^{2}x\sin{x}[/latex] and make the substitution [latex]{\sin}^{2}x=1-{\cos}^{2}x[/latex]. Thus,
In the next example, we see the strategy that must be applied when there are only even powers of [latex]\sin{x}[/latex] and [latex]\cos{x}[/latex]. For integrals of this type, the identities
are invaluable. These identities are sometimes known as power-reducing identities and they may be derived from the double-angle identity [latex]\cos\left(2x\right)={\cos}^{2}x-{\sin}^{2}x[/latex] and the Pythagorean identity [latex]{\cos}^{2}x+{\sin}^{2}x=1[/latex].
The general process for integrating products of powers of [latex]\sin{x}[/latex] and [latex]\cos{x}[/latex] is summarized in the following set of guidelines.
Problem-Solving Strategy: Integrating Products and Powers of [latex]\sin{x}[/latex] and [latex]\cos{x}[/latex]
If [latex]k[/latex] is odd, rewrite [latex]{\sin}^{k}x={\sin}^{k - 1}x\sin{x}[/latex] and use the identity [latex]{\sin}^{2}x=1-{\cos}^{2}x[/latex] to rewrite [latex]{\sin}^{k - 1}x[/latex] in terms of [latex]\cos{x}[/latex]. Integrate using the substitution [latex]u=\cos{x}[/latex]. This substitution makes [latex]du=\text{-}\sin{x}dx[/latex].
If [latex]j[/latex] is odd, rewrite [latex]{\cos}^{j}x={\cos}^{j - 1}x\cos{x}[/latex] and use the identity [latex]{\cos}^{2}x=1-{\sin}^{2}x[/latex] to rewrite [latex]{\cos}^{j - 1}x[/latex] in terms of [latex]\sin{x}[/latex]. Integrate using the substitution [latex]u=\sin{x}[/latex]. This substitution makes [latex]du=\cos{x}dx[/latex]. (Note: If both [latex]j[/latex] and [latex]k[/latex] are odd, either strategy 1 or strategy 2 may be used.)
If both [latex]j[/latex] and [latex]k[/latex] are even, use [latex]{\sin}^{2}x=\frac{1}{2}-\frac{1}{2}\cos\left(2x\right)[/latex] and [latex]{\cos}^{2}x=\frac{1}{2}+\frac{1}{2}\cos\left(2x\right)[/latex]. After applying these formulas, simplify and reapply strategies 1 through 3 as appropriate.
Since the power on [latex]\sin{x}[/latex] is even [latex]\left(k=4\right)[/latex] and the power on [latex]\cos{x}[/latex] is even [latex]\left(j=0\right)[/latex], we must use strategy 3. Thus,
Since [latex]{\cos}^{2}\left(2x\right)[/latex] has an even power, substitute [latex]{\cos}^{2}\left(2x\right)=\frac{1}{2}+\frac{1}{2}\cos\left(4x\right)\text{:}[/latex]
[latex]\begin{array}{cc}={\displaystyle\int}\left(\frac{3}{8}-\frac{1}{2}\cos\left(2x\right)+\frac{1}{8}\cos\left(4x\right)\right)dx\hfill & \text{Simplify}.\hfill \\ =\frac{3}{8}x-\frac{1}{4}\sin\left(2x\right)+\frac{1}{32}\sin\left(4x\right)+C\hfill & \text{Evaluate the integral}.\hfill \end{array}[/latex]
In some areas of physics, such as quantum mechanics, signal processing, and the computation of Fourier series, it is often necessary to integrate products that include [latex]\sin\left(ax\right)[/latex], [latex]\sin\left(bx\right)[/latex], [latex]\cos\left(ax\right)[/latex], and [latex]\cos\left(bx\right)[/latex]. These integrals are evaluated by applying trigonometric identities, as outlined in the following rule.
integrating products of aines and cosines of different angles
To integrate products involving [latex]\sin\left(ax\right)[/latex], [latex]\sin\left(bx\right)[/latex], [latex]\cos\left(ax\right)[/latex], and [latex]\cos\left(bx\right)[/latex], use the substitutions