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PROGRAMME 17. REDUCTION FORMULAS. Generating a reduction formula Definite integrals Integrands of the form and. Programme 17: Reduction formulas. Programme 17: Reduction formulas. Generating a reduction formula Definite integrals Integrands of the form and.

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PROGRAMME 17

REDUCTION FORMULAS


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Generating a reduction formula

Definite integrals

Integrands of the form and

Programme 17: Reduction formulas


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Programme 17: Reduction formulas

Generating a reduction formula

Definite integrals

Integrands of the form and


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Programme 17: Reduction formulas

Generating a reduction formula

Using the integration by parts formula:

it is easily shown that:


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Programme 17: Reduction formulas

Generating a reduction formula

Writing:

then

can be written as:

This is an example of a reduction formula.


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Programme 17: Reduction formulas

Generating a reduction formula

Sometimes integration by parts has to be repeated to obtain the reduction formula. For example:


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Programme 17: Reduction formulas

Generating a reduction formula

Definite integrals

Integrands of the form and


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Programme 17: Reduction formulas

Definite integrals

When the integral has limits the reduction formula may be simpler. For example:


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Generating a reduction formula

Definite integrals

Integrands of the form and

Programme 17: Reduction formulas


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Integrands of the form and

Programme 17: Reduction formulas

The reduction formula for is

and . . .


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Integrands of the form and

Programme 17: Reduction formulas

the reduction formula for is:

These take interesting forms when evaluated as definite integrals between 0 and /2


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Integrands of the form and

Programme 17: Reduction formulas

  • The reduction formulas for are both:

  • where

  • If n is even, the formula eventually reduces to I0 = /2

  • If n is odd the formula eventually reduces to I1 = 1


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Programme 17: Reduction formulas

Learning outcomes

  • Integrate by parts and generate a reduction formula

  • Integrate by parts using a reduction formula

  • Evaluate integrals with integrands of the form sinnx and cosnx using reduction formulas


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