ntegrals Involving Trig Functions

ntegrals Involving Trig Functions - ntegrals Involving Trig...

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ntegrals Involving Trig Functions In this section we are going to look at quite a few integrals involving trig functions and some of the techniques we can use to help us evaluate them. Let’s start off with an integral that we should already be able to do. This integral is easy to do with a substitution because the presence of the cosine, however, what about the following integral. Example 1 Evaluate the following integral. Solution This integral no longer has the cosine in it that would allow us to use the substitution that we used above. Therefore, that substitution won’t work and we are going to have to find another way of doing this integral. Let’s first notice that we could write the integral as follows, Now recall the trig identity,
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With this identity the integral can be written as, and we can now use the substitution . Doing this gives us, So, with a little rewriting on the integrand we were able to reduce this to a fairly simple substitution. Notice that we were able to do the rewrite that we did in the previous example because the exponent on the sine was odd. In these cases all that we need to do is strip out one of the sines. The exponent on the remaining sines will then be even and we can easily convert the remaining sines to cosines using the identity, (1) If the exponent on the sines had been even this would have been difficult to do. We could strip out a sine, but the remaining sines would then have an odd exponent and while we could convert them to cosines the resulting integral would often be even more difficult than the original integral in most cases. Let’s take a look at another example.
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Example 2 Evaluate the following integral. Solution So, in this case we’ve got both sines and cosines in the problem and in this case the exponent on the sine is even while the exponent on the cosine is odd. So, we can use a similar technique in this integral. This time we’ll strip out a cosine and convert the rest to sines. At this point let’s pause for a second to summarize what we’ve learned so far about integrating powers of sine and cosine. (2) In this integral if the exponent on the sines ( n ) is odd we can strip out one sine, convert the rest to cosines using (1) and then use the substitution
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. Likewise, if the exponent on the cosines ( m ) is odd we can strip out one cosine and convert the rest to sines and the use the substitution . Of course, if both exponents are odd then we can use either method. However, in these cases it’s usually easier to convert the term with the smaller exponent. The one case we haven’t looked at is what happens if both of the exponents are even? In this case the technique we used in the first couple of examples simply won’t work and in fact there really isn’t any one set method for doing these integrals. Each integral is different and in some cases there will be more than one way to do the integral. With that being said most, if not all, of integrals involving products of sines and
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ntegrals Involving Trig Functions - ntegrals Involving Trig...

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