4.7 Integrating Trigonometric Functions

4.7.1 Integrals of the form where m and n are integers.

1) If m = 2k + 1 is an odd positive integer, then set

Also do this if n is an odd positive integer.

Example 1. ]

2) If m and n are even positive integers, then the integrand (1) is transformed by means of

Example 2.

3) If m = -m and n = -n are negative integers of identical parity, then

In particular, the following integrals reduce to this case:

Example 3.

Example 4.

4) Integrals of the form

where m is a positive integer, are evaluated using

Example 5.

In the general case, integrals Im,n are evaluated by means of reduction formulae which, as a rule, are derived by integration by parts.

Example 6.

EXERCISES 1338 - 1364

Find the integrals

Hints and Answers 1338 - 1364

4.7.2 Integrals of the form In these cases, one employs the formulae:

1) sin mx cos nx = ˝[sin (m + n)x + sin (m - n)x],
2) sin mx sin nx = ˝[cos(m - n)x = cos(m + n)x,
3) cos mx cos nx = ˝(cos(m - n)x + cos(m+n)x].

Example 7.

EXERCISES 1365 - 1372

Hints and Answers 1365 - 1372

4.7.3 Integrals of the form
where R is a rational function.

1) Use

whence

and such integrals are reduced to integrals of rational functions in the new variable t.

Example 8. Find

Solution: Setting tan x/2 = t, we find

2) If we have the identity

we can use the substitution tan x = t to give the integral to a rational form.

Here,

Example 9. Find

Solution: Setting

we find

Note that the integral of Example 9. is evaluated faster, if the numerator and denominator of the fraction are first divided by cos˛x.

In individual cases, it is useful to apply artificial procedures (for example, Exercise 1379).

EXERCISES 1373 - 1390

Find the integrals:

Hints and Answers 1391 - 1402

4.8 Integration of Hyperbolic Functions

The integration of hyperbolic functions is completely analogous to that of the of trigonometric functions. The reader should recall the basic formulae:

Example 1. Find

Solution: We have

Example 2. Find

Solution: We have

EXERCISES 1391 - 1402

Hints and Answers 1393 - 1402

4.9 Trigonometric and Hyperbolic Substitutions for Integrals of the Form
where R is a rational function.

Transforming the quadratic ax˛ + bx + c into a sum or difference of squares, the integral becomes reducible to one of the types of integrals:

which are evaluated using the substitutions:

Example 1. Find

Solution: We have

Setting x + l = tan z, we have dx = sec˛z and

Example 2. Find

Solution: We have

Setting

we get

Since

we have finally

EXERCISES 1403 - 1414

Find the integrals:

Hints and Answers 1403 - 1414

4.10 Integration of Various Transcendental Functions

EXERCISES 1415 - 1426

1415 - 1426

4.11. Using Reduction Formulae

EXERCISES 1427 - 1430

Derive the reduction formulae for the integrals:

Hints and Answers 1427 - 1430

4.12 Miscellaneous Integration Examples

EXERCISES 1431 - 1500

Hints and Answers 1431 - 1500