4.5 Integration By Pattern Recognition A Mathematics Academy Production

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

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

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,

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.

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

Integrate Sub to get Integrate Back Substitute

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

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

Basic Trig Integrals

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!

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

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

Symmetry in Definite Integral Integrals of Symmetric Functions