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4.5 Basic Integration Formulas

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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,

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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

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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

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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

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Exercises: Perform the following integration

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Homework:

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

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