#
Integrating Powers and Product of Sines and Cosines

These are integrals of the following form:

We have two cases: both m and n are even or at least one of them is
odd.

**Case I: m or n odd**

Suppose *n* is odd. Hence *n* = 2*k* +
1. So
hold. Therefore, we have

which suggests the substitution . Indeed, we have and hence

The latest integral is a polynomial function of *u* which is easy to
integrate.

**Remark.** Note that if *m* is odd, then we
will split and carry the same calculations. In this case,
the substitution will be .

Example
1

**Case II: m and n are even**

The main idea behind is to use the trigonometric identities

Example
2

**Remark.** The following two formulas may be helpful in integrating powers of sine and cosine.

More Examples

More Challenging Problems

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**
[Calculus]
**** **
[Geometry]
[Algebra]
[Trigonometry ]
[Differential Equations]
[Complex Variables]
[Matrix Algebra]

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*
Author: Mohamed
Amine Khamsi *

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