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**4.5 Integration By Pattern Recognition**

A Mathematics Academy Production

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**Integration by Pattern Recognition:**

The first basic type of integration problem is in the form:

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**Note: If Integrate by recognizing the Pattern Then Integrating we get:**

Therefore, this integral is of the type: Substitute, But, Henceforth,

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**Note: This is exactly in the form!**

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

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**Note: This is not exactly in the form!**

Multiplying by a Form of 1 to integrate: Then Note: If 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.

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**Note: This is exactly in the form!**

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

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Integrate Sub to get Integrate Back Substitute

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

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**Trig Integrals in the form:**

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

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Basic Trig Integrals

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**The key to each basic Trig Integral is that:**

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

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**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

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Let u = x3 ; du = 3x2dx ; C.F. 1/3

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**Symmetry in Definite Integral**

Integrals of Symmetric Functions

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Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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