Integrating Products and Powers of sinx and cosx

Learning Outcomes

  • Solve integration problems involving products and powers of [latex]\sin{x}[/latex] and [latex]\cos{x}[/latex]

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 the following examples.

Example: Integrating [latex]{\displaystyle\int}{\cos}^{j}x\sin{x}dx[/latex]

Evaluate [latex]{\displaystyle\int}{\cos}^{3}x\sin{x}dx[/latex].

try it

Evaluate [latex]{\displaystyle\int}{\sin}^{4}x\cos{x}dx[/latex].

Watch the following video to see the worked solution to the above Try It

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You can view the transcript for this segmented clip of “3.2 Trigonometric Integrals” here (opens in new window).

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. We briefly review the relationships between these functions below.

Recall: The Pythagorean Identity

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:

[latex]\sin^2 x = 1 -\cos^2 x[/latex]

Example: Integrating [latex]{\displaystyle\int}{\cos}^{j}x{\sin}^{k}xdx[/latex] Where k is Odd

Evaluate [latex]{\displaystyle\int}{\cos}^{2}x{\sin}^{3}xdx[/latex].

try it

Evaluate [latex]{\displaystyle\int}{\cos}^{3}x{\sin}^{2}xdx[/latex].

Watch the following video to see the worked solution to the above Try It

.

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.

You can view the transcript for this segmented clip of “3.2 Trigonometric Integrals” here (opens in new window).

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

[latex]{\sin}^{2}x=\frac{1}{2}-\frac{1}{2}\cos\left(2x\right)=\frac{1-\cos\left(2x\right)}{2}[/latex]

 

and

[latex]{\cos}^{2}x=\frac{1}{2}+\frac{1}{2}\cos\left(2x\right)=\frac{1+\cos\left(2x\right)}{2}[/latex]

 

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].

example: Integrating an Even Power of [latex]\sin{x}[/latex]

Evaluate [latex]{\displaystyle\int}{\sin}^{2}xdx[/latex].

try it

Evaluate [latex]\displaystyle\int {\cos}^{2}xdx[/latex].

Try It

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 sin x and cos x

To integrate [latex]{\displaystyle\int}{\cos}^{j}x{\sin}^{k}xdx[/latex] use the following strategies:

  1. 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].
  2. 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.)
  3. 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.

Example: Integrating [latex]\displaystyle\int {\cos}^{j}x{\sin}^{k}xdx[/latex] where k is Odd

Evaluate [latex]{\displaystyle\int}{\cos}^{8}x{\sin}^{5}xdx[/latex].

Example: Integrating [latex]{\displaystyle\int}{\cos}^{j}x{\sin}^{k}xdx[/latex] where k and j are Even

Evaluate [latex]{\displaystyle\int}{\sin}^{4}xdx[/latex].

try it

Evaluate [latex]{\displaystyle\int}{\cos}^{3}xdx[/latex].

try it

Evaluate [latex]{\displaystyle\int}{\cos}^{2}\left(3x\right)dx[/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.

Rule: Integrating Products of Sines 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

[latex]\sin\left(ax\right)\sin\left(bx\right)=\frac{1}{2}\cos\left(\left(a-b\right)x\right)-\frac{1}{2}\cos\left(\left(a+b\right)x\right)[/latex]
[latex]\sin\left(ax\right)\cos\left(bx\right)=\frac{1}{2}\sin\left(\left(a-b\right)x\right)+\frac{1}{2}\sin\left(\left(a+b\right)x\right)[/latex]
[latex]\cos\left(ax\right)\cos\left(bx\right)=\frac{1}{2}\cos\left(\left(a-b\right)x\right)+\frac{1}{2}\cos\left(\left(a+b\right)x\right)[/latex]

These formulas may be derived from the sum-of-angle formulas for sine and cosine.

Example: Evaluating [latex]\displaystyle\int \sin\left(ax\right)\cos\left(bx\right)dx[/latex]

Evaluate [latex]{\displaystyle\int}\sin\left(5x\right)\cos\left(3x\right)dx[/latex].

try it

Evaluate [latex]{\displaystyle\int}\cos\left(6x\right)\cos\left(5x\right)dx[/latex].