Trigonometric Integrals (Level 6) In the previous video we covered 4 distinct
cases of trigonometric integrals whose integrand contained powers of sine and cosine. Specifically
integrals of the form sine raised to the power of m of x times cosine raised to
the power of n of x dx. It is important to note that all the integrals from the previous 5
videos, contained the same expression inside the argument of the trigonometric function
in other words both sine and cosine contained the same angle inside the argument. At times
the process of reducing integrals in which both the powers on sine and cosine are even
tends to produce products of sine and cosine whose arguments are different. When you are
face to face with products of sine and cosine of the form sine of mx times cosine of nx
dx and sine of mx times sine of nx dx and cosine of mx times cosine of nx dx.
You want to make use of the following product to sum identities ½ times the quantity sine
of A minus B plus sine of A plus B and ½ times the quantity cosine of A minus B minus
cosine of A plus B and ½ times the quantity cosine of A minus B plus cosine of A plus
B. Respectively, these same expressions can be written in terms of the constants m and
n as follows. By making use of these identities we can reduce the products of sine and cosine
into sums of sine and cosine whose integral is relatively easier to find. Before we go
over some examples I want you to notice a couple of things. First, notice that these
identities can only be applied when both sine and cosine are linear functions in other words
the highest degree or exponent of the trigonometric functions has to equal 1. Second, the product
sine times cosine is not the only combination that can potentially have distinct angles,
products of sine and cosine can also occur so be on the lookout for products with the
same trigonometric functions but whose argument contain distinct angles. With that said, let’s
go over the first example. Find the integral of sine of 5x times sine
of 7x dx. Notice that both trigonometric functions are
linear meaning they have a degree of 1 their order is equal to 1. In addition, they contain
distinct arguments or in this case distinct angles. One is 5x and the other is 7x.
It is very important that you are able to spot these distinctions you really don’t want
to find the integral by using the techniques from the previous video they will most likley lead
you nowhere. The only exception is that you can actually solve this integral by using
integration by parts, if you want a challenge I invite you to try this integral by using
integration by parts. This integral can be solved a lot easier by using the following
product to sum identities. Remember there are 3 different identities depending of what
combination of trigonometric functions we have in this case we have sine times sine
they also contain different arguments. This means that we need to make use of the following
product to sum identity. With this identity products of sines can be written into sums
of cosines, for this integral A is equal to 5x and B is equal to 7x. So we go ahead and
subtract the angles on the first expression and add the angles in the second expression
doing that we obtain the following. Next we go ahead and factor out the constant ½, notice
that we now have a sum and or difference of trigonometric functions, now we can find the
integral term by term, the integral of the first expression requires a u-substitution,
which results in the following integral, the second expression also requires a u-substitution
which results in the following integral, then we make sure we include the constant C, the
final step is to distribute the constant ½ doing that we obtain the final answer equal
to sine of 2x over 4 minus sine of 12x over 24 plus C. These integrals are relatively easier
than the integrals from the previous videos the only hard part was to make sure we identify
when we need to use the product to sum identities. Alright let’s try the next example.
Find the integral of cosine of 4x times cosine of 9x dx.
Once again notice that we have products of trigonometric functions of degree 1 the highest
exponent on each function is equal to 1. In, addition both functions contain different
expressions inside their respective arguments one is 4x and the other one is 9x. This means
that we need to use a product to sum identity, notice that this integral contains a product
of cosines so we need to make use of the following product to sum identity. Notice that this
identity looks similar to the previous product to sum identity, both expressions contain
cosines, and the arguments are also the same the only distinction is that this one contains
a sum and other identity contains a difference. So using 4x as our A and 9x as our B we go
ahead and subtract them in the first expression, then we go ahead and add them in the second
expression. Doing that we obtain the following expression. Next we go ahead and factor out
the constant 1/2, then it’s just matter of finding the integral term by term.
The integral of the first expression requires a u-substitution, this results in the following
integral, then we go ahead and find the integral of the second expression this one also requires
a u-substitution finding the integral results in the following expression. Having found the integral
of each term we go ahead and add the constant C, the final step is to distribute the constant
½ doing that we obtain the final answer equal to sine of 5x over 10 plus sine of 13x over
26 plus C. Make sure you identify the type of product that you are face to face with,
this way you can use the correct product to sum identity to find the integral. Alright
let’s end the video with the final example.
Find the integral of sine of pi x times cosine of 3 pi x dx.
Alright here we have a product of sin and cosine each having a degree of 1, they also contain
distinct arguments this means that we need to use a product to sum identity, we need
to use the identity that contains sine and cosine, so we make use the following identity.
Using pi x as our A and 3 pi x as our B we go ahead and rewrite this expression into
sums of sines, for the first expression we go ahead and subtract the angles, and for
the second expression we go ahead and add the angles. Doing that we obtain the following expression.
Then we go ahead and factor out the constant 1/2 , now that we have sums of sines
we go ahead and take the integral of each expression, the integral of the first expression
requires a u-substitution, which result in the following integral, the second expression
also requires a u-substitution and this one results in the following integral, then we
go ahead and include the constant C. The next step is to distribute the constant ½ doing that
we obtain the following. Notice that one of our expressions contains a negative angle
inside of the argument, for the most part we don’t want to have negative arguments
we can fix this by remembering that cosine is technically an even function this means
that it posses the following property, as a consequence this negative angle results
in the same expression as if it were a positive angle, so we might as well rewrite it with
a positive argument. A note of caution this is not the case with sine, sine is technically
an odd function and posses the following property. Remember if you still need a list of trigonometric
identities you can go to Math Fortress.com and download a trigonometric identities formula
sheet in the precalculus section. So rewriting the negative argument results in the final
expression equal to cosine of 2 pi x over 4 pi minus cosine of 4 pi x over 8 pi plus C.
Alright, at this point we’ve covered pretty
much all the possible cases involving products of sine and cosine. In our next video we
will go over special integrals that contain combinations of sine and cosine that do not
fit nicely into one of the forms that we have seen so far, nevertheless all the techniques that we have
learned up to this point can be used to find the integral.