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Generating Reduction Formulas for Definite Integrals Using Integration by Parts

This program focuses on generating reduction formulas for definite integrals with integrands of specific forms. It explores how integration by parts can be utilized to derive these formulas, sometimes requiring repetition to achieve the desired result. The reduction formulas adapt based on whether the integral's parameter is even or odd, simplifying evaluation between limits 0 and ( frac{pi}{2} ). Key learning outcomes include integrating by parts, generating reduction formulas, and evaluating integrals involving ( sin nx ) and ( cos nx ).

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Generating Reduction Formulas for Definite Integrals Using Integration by Parts

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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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