1. STROUD Worked examples and exercises are in the text Programme 18: Reduction formulas REDUCTION FORMULAS PROGRAMME 18

2. STROUD Worked examples and exercises are in the text Programme 18: Reduction formulas Generating a reduction formula Definite integrals Integrands of the form and

3. STROUD Worked examples and exercises are in the text Programme 18: Reduction formulas Generating a reduction formula Definite integrals Integrands of the form and

4. STROUD Worked examples and exercises are in the text Programme 18: Reduction formulas Generating a reduction formula Using the integration by parts formula: it is easily shown that:

5. STROUD Worked examples and exercises are in the text Programme 18: Reduction formulas Generating a reduction formula Writing: then can be written as: This is an example of a reduction formula.

6. STROUD Worked examples and exercises are in the text Programme 18: Reduction formulas Generating a reduction formula Sometimes integration by parts has to be repeated to obtain the reduction formula. For example:

7. STROUD Worked examples and exercises are in the text Programme 18: Reduction formulas Generating a reduction formula Definite integrals Integrands of the form and

8. STROUD Worked examples and exercises are in the text Programme 18: Reduction formulas Generating a reduction formula Definite integrals Integrands of the form and

9. STROUD Worked examples and exercises are in the text Programme 18: Reduction formulas Definite integrals When the integral has limits the reduction formula may be simpler. For example:

10. STROUD Worked examples and exercises are in the text Programme 18: Reduction formulas Generating a reduction formula Definite integrals Integrands of the form and

11. STROUD Worked examples and exercises are in the text Programme 18: Reduction formulas Generating a reduction formula Definite integrals Integrands of the form and

12. STROUD Worked examples and exercises are in the text Programme 18: Reduction formulas Integrands of the form and The reduction formula for is and...

13. STROUD Worked examples and exercises are in the text Programme 18: Reduction formulas the reduction formula for is: These take interesting forms when evaluated as definite integrals between 0 and π/2 Integrands of the form and

14. STROUD Worked examples and exercises are in the text Programme 18: Reduction formulas Integrands of the form and The reduction formulas for are both: where (a)If n is even, the formula eventually reduces to I 0 = π/2 (b)If n is odd the formula eventually reduces to I 1 = 1

15. STROUD Worked examples and exercises are in the text Programme 18: 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 sin n x and cos n x using reduction formulas