1. 4.5 Integration By Pattern Recognition
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2. Integration by Pattern Recognition:
The first basic type of integration
problem is in the form:

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

4. 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,

5. 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.

6. 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

7. Integrate
Sub to get
Integrate
Back Substitute

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

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

10. Basic Trig Integrals

11. 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!

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

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

14. Symmetry in Definite Integral
Integrals of Symmetric Functions