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PROGRAMME 17

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

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  1. PROGRAMME 17 REDUCTION FORMULAS

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

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

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

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

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

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

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

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

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

  11. 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

  12. 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

  13. 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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