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4.5 Integration By Pattern Recognition A Mathematics Academy Production.

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Presentation on theme: "4.5 Integration By Pattern Recognition A Mathematics Academy Production."— Presentation transcript:

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2 4.5 Integration By Pattern Recognition A Mathematics Academy Production

3 Integration by Pattern Recognition: The first basic type of integration problem is in the form:

4 Integrate by recognizing the Pattern Then Therefore, this integral is of the type: Note: If Integrating we get: Substitute, Henceforth, But,

5 Note: If Then Note: This is exactly in the form! Therefore, this integral is of the type: Integrating we get: But, Substitute, Henceforth,

6 Multiplying by a Form of 1 to integrate: Then Note: This is not exactly in the form! The inside of the Integral has to be multiplied by 2 Therefore the outside of the Integral has to be multiplied by ½, since( 2) (½) = 1, and as long as we multiple the entire integral by a numeric form of 1 we can proceed with integration. Note: If

7 Now multiply by a form of 1 to integrate: Then Substitute get Note: This is exactly in the form! Integrate this form to get Simplifying to get Note: If

8 Integrate Sub to getIntegrateBack Substitute

9 Ex. Evaluate Sub in Integrate Pick u, compute du Sub in

10 9 Trig Integrals in the form: Let Then Note: This is exactly in the form! Integrate this form to get Sub in

11 10 Basic Trig Integrals

12 The key to each basic Trig Integral is that: Let u = The angle While du = The derivative of the angle First make sure you do not have a problem. You need to know the 6 trig. Derivatives, so that you can work backwards and find their Anti-derivatives!

13 12 Using the Trig Integrals The technique is often to find a u which is the angle, the argument of the trig function Consider What is the u, the du? Substitute, integrate

14 Let u = x 3 ; du = 3x 2 dx; C.F. 1/3

15 Symmetry in Definite Integral Integrals of Symmetric Functions


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