(In what follows, I will write for and for )
The formulas below make use of these identities repeatedly.
We want a series of methods and reduction formulas that allow us to evaluate any expression of the form
for and integers,
1. Integrating powers of
In lecture we saw that
and derived the following reduction formula for integrating powers of
This formula is valid for any
(The integral of was obtained by what is essentially the method of partial fractions, that we will study in detail when we reach section 7.4 in the book. The reduction formula was obtained using integration by parts.)
Exercise 1 Find and
2. Integrating powers of
We also saw that
and derived the following reduction formula:
valid for all
3. Integrating products of powers of and
Suppose now that we need to evaluate an expression of the form
where both and are at least 1. As in the case of integrals of products of powers of sines and cosines, it is best to divide the problem into two cases.
3.1. If is odd
Suppose first that is odd, say for some integer Then
Since we can write The last integral can then be expressed in the form
This can be easily evaluated using the substitution that transforms it into
To evaluate this expression, expand and multiply the result by This gives us a polynomial in We can integrate the polynomial term by term, and then replace back in place of
Exercise 3 Find and
3.2. If is even
Suppose now that is even, say for some integer Then
To evaluate this expression, expand and multiply the result by This gives us a sum of powers of that can be evaluated term by term using the reduction formula from Section 1. Note that this method works even if
Exercise 4 Find
Exercise 5 Find using this method, and show that your answer actually gives the same result as the answer you found in Exercise 2.
4. Integrating powers of and
Exercise 6 (This is long.) Explain how to adapt the methods from the previous sections to find any integral of the form Then repeat the previous exercises but with in place of and instead of
Although I am not writing this as an exercise, it is a good idea to also spend some time thinking about what one would do with integrals of products of powers of and or of and or of all four expressions.
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