where m
and n are integers. 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.
Find the integrals
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.
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).
Find the integrals:
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
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
Find the integrals:
4.10 Integration of Various Transcendental Functions
4.11. Using Reduction Formulae
Derive the reduction formulae for the integrals:
4.12 Miscellaneous Integration Examples
