Integration of Trigonometric Polynomials

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Integration of Trigonometric Polynomials
Integrals of the type
Case 1 n is odd Case 2 m is odd Case 3
both m and n are even General Trigonometric
Polynomials
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Basic Trigonometric Monomials 1
Problem
Example
This computation was possible since the function
to be integrated was a (simple) polynomial in the
sine function times the cosine function.
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Basic Trigonometric Monomials 2
Problem
Case 1 n is odd.
If n 2k1 is odd then one can rewrite the
function to be integrated using the formula
This can be reduced to an integral of a
polynomial by the substitution t sin(x).
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Basic Trigonometric Monomials 3
Example
Solution
Rewrite
The last integral is easy to compute. One gets
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Basic Trigonometric Monomials 4
Problem
Case 2 m is odd
If m 2k1 is odd then one can rewrite the
function to be integrated using the formula
This can be reduced to an integral of a
polynomial by the substitution t cos(x).
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Basic Trigonometric Monomials 5
Example
Solution
Rewrite
The last integral is easy to compute. One gets
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Basic Trigonometric Monomials 6
Problem
Case 3 m and n even
If both m and n are even use the
trigonometric formulae
to simplify the function to be integrated.
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Basic Trigonometric Monomials 7
Example
Solution
Now use the same trig formula again!
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Integration of Trigonometric Polynomials
Problem
Solution