PROGRAMME 17 REDUCTION FORMULAS.

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Presentation transcript:

PROGRAMME 17 REDUCTION FORMULAS

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

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

Programme 17: Reduction formulas Generating a reduction formula Using the integration by parts formula: it is easily shown that:

Programme 17: Reduction formulas Generating a reduction formula Writing: then can be written as: This is an example of a reduction formula.

Programme 17: Reduction formulas Generating a reduction formula Sometimes integration by parts has to be repeated to obtain the reduction formula. For example:

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

Programme 17: Reduction formulas Definite integrals When the integral has limits the reduction formula may be simpler. For example:

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

Programme 17: Reduction formulas Integrands of the form and The reduction formula for is and . . .

Programme 17: Reduction formulas Integrands of the form and the reduction formula for is: These take interesting forms when evaluated as definite integrals between 0 and /2

Programme 17: Reduction formulas Integrands of the form and 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

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