Integration Techniques Kunal, JC, and Varun

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.

29. Improper Integrals

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

31. Trig Rules For Integration (+c)

32. Inverse Trig Rules for Integration (+c)

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

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

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 + 1 + 4 x2 + 2x + 1 + 4 = (x + 1)2 + 4 =

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

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.

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.

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 ]

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.

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

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