4.5 Basic Integration Formulas

1. 4.5 Basic Integration Formulas Agenda: develop techniques for integrating algebraic functions. Using new notation, the formula for the antiderivative of a power function can be rewritten as Constant Multiplier Integration Rule: Proof: This integral is an _________ of ___. Take the differential of it: by the definition, . But . Thus, 1

2. Proof (cntd): By the constant multiplier (differentiation) rule: and the equality at the end of the last slide takes the form Next, we take the indefinite integral (antiderivative) of that: By the definition of the integral, , we get 2

3. We can combine the above constant multiplier integration rule and the sum integration rule (the integral of a sum is equal to the sum of integrals) to form the following Linear Combination Rule: Next,we try to reverse the generalized rules obtained using the chain rule: Example: Differentiate the function and try to formulate the integration rule corresponding to the reverse operation. Solution: Then, differential writes as Now, we know that the function F(x) itself is the antiderivative of its derivative, and it writes as 3

4. Example (ctnd): Try to integrate this same function straightforward: Since 2xdx=d(x2), Next, we can take out the constant multiplier 2 and split the integral into the sum of integrals, but there is another, more elegant way. Observe that for any constant c. In our case, we use c=1 and write Next, we denote the function under the differential as a new variable u and the integral becomes . The final step is substituting u as a function of x back 4

5. Exercises: Perform the following integration 5

6. Homework: Section 4.5: 1,5,7,9,13,15,17,21,23,25,49,61,67. 6