1. Integration Techniques
Kunal, JC, and Varun

2. Integration
Definition: To put together parts or elements and combine them into a whole
Historical: To put together the races
Calculus: a way of adding slices to find the whole, most commonly this whole is the area under a curve. Integrating is just adding up an amount of slices of Δx that approaches infinity under the curve.

3. 29. Improper Integrals

4. 30. Power Rule for Integration (+c)
Add 1 to the exponent
Divide term by the exponent+1
Don’t forget to add + C

5. 31. Trig Rules For Integration (+c)

6. 32. Inverse Trig Rules for Integration (+c)

7. 33. Exponential and Log Rules for Integration (+c)1

8. 34. U-Substitution Define U Differentiate U Solve
Substitute U and DU into the integral
Integrate
Replace U with X
Solve for final answer

9. 35. Integration using Completing the Square
Complete the square to make the expression in the denominator factorable
Factor first three terms
Substitute back into integral
Substitute using a trig rule
x2 + 2x
x2 + 2x = (x + 1)2 + 4 =

10. 36. Integration by Parts Separate the integral into u and dv
Find v by integrating dv
Substitute into
*for determining u

11. 37. Integration with Partial Functions
Factor the denominator
Set the function equal to “A” over one factor of the denominator and “B” over the other factor.
Multiply the whole equation by the denominator
Set x equal to such a value that A is multiplied by 0 to find B and vice versa to find A.
Substitute the calculated values of A and B back into the original equation where the denominator was separated and integrate both.

12. 38. Integrating Absolute Value Functions
If you cannot integrate I f(x) I by graphing, integrate algebraically
Split up the integral where f(x)=0
Test on which interval the graph is positive and integrate f(x) there. Test where the graph is negative and integrate -f(x) there.

13. 39. Limit Definition of a Definite Integral
If f is defined on a to b the definite integral on this interval is
Where Δx is (b-a)/n and for each i, xi = a + i∆x, and ci is a point in [xi−1 , xi ]

14. 40. Fundamental Theorem of Calculus Part 1 (Evaluating a definite integral)
Integrate the function
Substitute the lower and higher endpoint to the interval into the antiderivative.
Subtract the the value yielded from substituting the lower endpoint from that of the higher endpoint and solve.

15. 41. Fundamental Theorem of Calculus Part 2 (derivative of an integral)
Substitute the higher limit into the function and multiply by the derivative of that limit.
Substitute the lower limit into the function and multiply by the derivative of that limit.
Step 1 - step 2

16. Homework Problems Integration Techniques AB: 9-11, 13, 14, 17-23
Integration Techniques BC: 1-3, 6, 7, 13-17, 22, 23