Other Trigonometric Powers

These are integrals of the form

displaymath58

In every single one of these integrals, we will develop what is commonly called a Reduction Formula. The main idea behind is a smart use of trigonometric identities. Let us describe how it works.

Let us show how one can generate a reduction formula for tex2html_wrap_inline60 . The other once, will be given without any proof. We have

displaymath78

Since the derivative of tex2html_wrap_inline80 is tex2html_wrap_inline82 , we get

displaymath84

Therefore, we have

displaymath86

This is the reduction formula associated to the tangent function. What it says is that in order to find the integral of tex2html_wrap_inline60 it is enough to find the integral of tex2html_wrap_inline90 . This way, we can reduce the power n all the way down to 1 or 0. Recall that tex2html_wrap_inline94 . Let us give a table for all the reduction formulas.

displaymath96

where a is an arbitrary constant and tex2html_wrap_inline100 .

Example 1

Remark. Note that for tex2html_wrap_inline102 and tex2html_wrap_inline104 when n is even can be handled in a easier way. Indeed, we have

displaymath108

which suggests the substitution tex2html_wrap_inline110 . The same idea works for the cosecant function (in this case, the substitution will be tex2html_wrap_inline112 ).

Example 2

More Examples

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Author: Mohamed Amine Khamsi

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