**4.7
Integrating Trigonometric Functions**

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

1) If *m = *2*k +* 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 *I*_{m,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*˛ + *b*x
+ *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: