175 – Integrating products of secants and tangents

(In what follows, I will write for and for )

Recall that

that

and that

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

Exercise 2 Find and

 

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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