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:
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 .
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
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
Try to integrate this same function straightforward:
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
Exercises: Perform the following integration
Section 4.5: 1,5,7,9,13,15,17,21,23,25,49,61,67.