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Notes for Math 220, Calculus 2

Brenton LeMesurier

Section 3.2 Trigonometric Integrals

References.

Introduction.

One recurring challenge with integration is handling products. Integration by Parts helps with many products of common elementary functions; here we learn how to deal with products and powers of trigonometric functions, with the most fundamental cases being products of sines and cosines.
Some key tools for “eliminating products” are
Integration by Substitution
\begin{equation*} \int f(u(x)) \frac{du}{dx} \, dx = \int f(u)\, du \end{equation*}
The Trig. Identities
\begin{align} \cos^2(ax) \amp= \frac12 [1 + \cos(2ax)]\tag{3.2.1}\\ \sin^2(ax) \amp= \frac12 [1 - \cos(2ax)]\tag{3.2.2}\\ \sin(x) \cos(x) \amp= \frac12 \sin(2x)\tag{3.2.3} \end{align}
Note that in each case above, the substitution is the complementary function to the one appearing as an odd power.
If instead both sine and cosine appear as even powers, we can convert to a case with one of them an odd power by (possibly repeated) use of the Half-Angle Formulas; using these identities lowers the powers of the trig. functions in the formula, so if used often enough one power must become odd, and then the previous substitution strategy works.

Strategy for Evaluating \(\int \sin^m (ax) \cos^n (ax) \, dx\) (for \(m,n \geq 0\) mostly).

  1. If the power of cosine is odd and positive, use
    \begin{equation*} \cos^2(ax) = 1 - \sin^2(ax) \end{equation*}
    to reduce to a single factor of \(\cos(a x)\text{,}\) and then use the substitution
    \begin{equation*} u=\sin(ax), \, du=a \cos(ax) \, dx. \end{equation*}
  2. If the power of sine is odd and positive, use
    \begin{equation*} \sin^2(ax) = 1 - \cos^2(ax) \end{equation*}
    to reduce to a single factor of \(\sin(ax)\text{,}\) and then use the substitution
    \begin{equation*} u=\cos(ax), \, du=-a \sin(ax) \, dx. \end{equation*}
  3. If both powers are even and non-negative, use the half-angle identities (3.2.1) and (3.2.2) to half both powers (and double \(a\)).
    Repeat as necessary until one power is odd; then one can use method (a) or (b) above.
Sometimes it is also convenient to use the formula \(\sin(ax) \cos(ax) = \frac12 \sin(2ax)\) mentioned above.

Products with Other Trig. Functions: \(\tan x\text{,}\) \(\sec x\text{,}\) etc..

To integrate products involving the other four trig. functions, it sometimes works to express them in terms of sines and cosines; this works so long as you end up with an odd positive power of one of the two.

Products of Powers of \(\tan x\) and \(\sec x\).

If the above strategy fails because both powers are even and at least one is negative, it sometimes helps to rewrite as
\begin{equation*} \int \tan^j(ax) \sec^k (ax) \, dx. \end{equation*}
Then if the power \(k\) of secant is even and positive, \(k=2p+2\text{,}\) use the substitution
\begin{equation*} u=\tan(ax), \, du = a \sec^2(ax) dx \end{equation*}
by reserving a factor \(\sec^2(ax)\) for the differential and rewriting the remaining factor
\(\sec^{k-2}(ax)=[\sec^2(ax)]^p\) as \([1+\tan^2(ax)]^p.\)
Note that converting to \(\ds \int \frac{1}{\cos^4 x} dx\) is not so useful because the half-angle formula method then gives \(\ds \int \frac{4}{(1+\cos(2x))^2}\text{.}\)

Eliminating Products of Sines and Cosines.

Products of sines and cosines (possibly with different frequencies) can be eliminated in favor of sums, using the Product-to-Sum Identities for Trig. Functions:
\begin{align} \cos A \cos B \amp= \frac12 \left[ \cos(A-B) + \cos(A+B) \right]\tag{3.2.4}\\ \sin A \cos B \amp= \frac12 \left[ \sin(A-B) + \sin(A+B) \right]\tag{3.2.5}\\ \sin A \sin B \amp= \frac12 \left[ \cos(A-B) - \cos(A+B) \right]\tag{3.2.6} \end{align}

Other Cases: Some Useful Integrals.

Some cases are not covered by any of the above methods, and so require further experimentation with trig. identities, integration by parts and such.
Then sometimes the methods to be seen in Section 3.4 can help, when negative powers are present so that substitutions give rational functions instead of polynomials.
The most commonly encountered “outlier” is
\begin{equation} \int \sec x \, dx = \ln |\sec x + \tan x| + C\tag{3.2.7} \end{equation}
We will soon be able to evaluate this, using ideas from Section 3.3 and Section 3.4; meanwhile:

Section Study Guide.

Study Calculus Volume 2, Section 3.2 2 ; ou main focus is the first case of sine-cosisse products to in particular look at
  • The Problem Solving Stategy for sine-cosine products
  • Examples 8 to 13
  • Checkpoints 5 to 10
  • and work one or several exercises from each of the groups: 69–72, (73, 74 & 76), 79–86, 97–101, and 103–106.
openstax.org/books/calculus-volume-2/pages/3-2-trigonometric-integrals
openstax.org/books/calculus-volume-2/pages/3-2-trigonometric-integrals
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