Trigonometric functions are useful for describing periodic behavior. This section describes several techniques for finding antiderivatives of certain combinations of trigonometric functions.
In learning the technique of Substitution, we saw the integral in Example 5.5.4. The integration was not difficult, and one could easily evaluate the indefinite integral by letting or by letting . This integral is easy since the power of both sine and cosine is 1.
We generalize this and consider integrals of the form , where are nonnegative integers. Our strategy for evaluating these integrals is to use the identity to convert high powers of one trigonometric function into the other, leaving a single sine or cosine term in the integrand. We summarize the general technique in the following Key Idea.
Consider , where are nonnegative integers.
If is odd, then for some integer . Rewrite
where and .
If is odd, then using substitutions similar to that outlined above we have
where and .
If both and are even, use the half-angle identities
to reduce the degree of the integrand. Expand the result and apply the principles of this Key Idea again.
We practice applying Key Idea 8.2.1 in the next examples.
SolutionThe power of the sine factor is odd, so we rewrite as
Our integral is now . Let , hence . Making the substitution and expanding the integrand gives
SolutionBecause the powers of both the sine and cosine factors are odd, we can apply the techniques of Key Idea 8.2.1 to either power. We choose to work with the power of the sine factor since that has a smaller exponent.
We rewrite as
This lets us rewrite the integral as
Substituting and integrating with and , we have
Instead, another approach would be to rewrite as
We rewrite the integral as
Now substitute and integrate, using and .
The work we are doing here can be a bit tedious, but the skills developed (problem solving, algebraic manipulation, etc.) are important. Nowadays problems of this sort are often solved using a computer algebra system. The powerful program Mathematica® integrates as
which clearly has a different form than our second answer in Example 8.2.2, which is ††margin: Λ
Figure 8.2.1 shows a graph of and ; they are clearly not equal, but they differ only by a constant: for some constant . So we have two different antiderivatives of the same function, meaning both answers are correct.
SolutionThe power of sine is even so we employ a half-angle identity, algebra and a u- substitution as follows:
SolutionThe powers of sine and cosine are both even, so we employ the half-angle formulas and algebra as follows.
The term is easy to integrate. The term is another trigonometric integral with an even power, requiring the half-angle formula again. The term is a cosine function with an odd power, requiring a substitution as done before. We integrate each in turn below.
Finally, we rewrite as
Letting , we have , hence
Putting all the pieces together, we have
The process above was a bit long and tedious, but being able to work a problem such as this from start to finish is important.
When evaluating integrals of the form , the Pythagorean Theorem allowed us to convert even powers of sine into even powers of cosine, and vice versa. If, for instance, the power of sine was odd, we pulled out one and converted the remaining even power of into a function using powers of , leading to an easy substitution.
The same basic strategy applies to integrals of the form , albeit a bit more nuanced. The following three facts will prove useful:
(the Pythagorean Theorem).
If the integrand can be manipulated to separate a term with the remaining secant power even, or if a term can be separated with the remaining power even, the Pythagorean Theorem can be employed, leading to a simple substitution. This strategy is outlined in the following Key Idea.
Consider , where and are nonnegative integers.
If is even, then for some integer . Rewrite as
where and .
If is odd and , then for some integer . Rewrite as
where and .
If is odd and is even, then for some integer . Convert to . Expand the new integrand and use Integration By Parts, with .
If is even and , rewrite as
The techniques described in items 1 and 2 of Key Idea 8.2.2 are relatively straightforward, but the techniques in items 3 and 4 can be rather tedious. A few examples will help with these methods.
SolutionSince the power of secant is even, we use rule #1 from Key Idea 8.2.2 and pull out a in the integrand. We convert the remaining powers of secant into powers of tangent.
|Now substitute, with , with .|
|We leave the integration and subsequent substitution to the reader. The final answer is|
We derived integrals for tangent and secant in Section 5.5 and will regularly use them when evaluating integrals of the form . As a reminder:
SolutionWe apply rule #3 from Key Idea 8.2.2 as the power of secant is odd and the power of tangent is even (0 is an even number). We use Integration by Parts; the rule suggests letting , meaning that .
Employing Integration by Parts, we have
|This new integral also requires applying rule #3 of Key Idea 8.2.2:|
In previous applications of Integration by Parts, we have seen where the original integral has reappeared in our work. We resolve this by adding to both sides, giving:
We give one more example.
SolutionWe employ rule #4 of Key Idea 8.2.2.
|We integrate the first integral with substitution, and ; and the second by employing rule #4 again.|
|Again, use substitution for the first integral and rule #4 for the second.|
Not surprisingly, evaluating integrals of the form is similar to evaluating . The guidelines from Key Idea 8.2.2 and the following three facts will be useful:
SolutionSince the power of cosecant is even we will let and save a for the resulting .
The integration and substitution required to finish this example are similar to that of previous examples in this section. The result is
Functions that contain products of sines and cosines of differing periods are important in many applications including the analysis of sound waves. Integrals of the form
are best approached by first applying the Product to Sum Formulas of Trigonometry found in the back cover of this text, namely
SolutionThe application of the formula and subsequent integration are straightforward:
Combinations of trigonometric functions that we have not discussed in this chapter are evaluated by applying algebra, trigonometric identities and other integration strategies to create an equivalent integrand that we can evaluate. To evaluate “crazy” combinations, those not readily manipulated into a familiar form, one should use integral tables. A table of “common crazy” combinations can be found at the end of this text.
These latter examples were admittedly long, with repeated applications of the same rule. Try to not be overwhelmed by the length of the problem, but rather admire how robust this solution method is. A trigonometric function of a high power can be systematically reduced to trigonometric functions of lower powers until all antiderivatives can be computed.
The next section introduces an integration technique known as Trigonometric Substitution, a clever combination of Substitution and the Pythagorean Theorem.
T/F: cannot be evaluated using the techniques described in this section since both powers of and are even.
T/F: cannot be evaluated using the techniques described in this section since both powers of and are odd.
T/F: This section addresses how to evaluate indefinite integrals such as
T/F: Sometimes computer programs evaluate integrals involving trigonometric functions differently than one would using the techniques of this section. When this is the case, the techniques of this section have failed and one should only trust the answer given by the computer.
In Exercises 5–32., evaluate the indefinite integral.
In Exercises 33–40., evaluate the definite integral.
Find the area between the curves and on the interval .