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POWER OF A FUNCTION

The integral of a function raised to a power is found by the following steps:

1. Increase the power of the function by 1.

2. Divide the result of step 1 by this increased power. 3. Add the constant of integration.

Formula.

PROOF:

Therefore,

NOTE: Recall that

EXAMPLE: Evaluate

SOL UTION: Let

so that

or

Then by substitution,

Therefore,

When you use this formula the integral must contain precisely du. If the required constant in du is not present, it must be placed in the integral and then compensation must be made.

EXAMPLE: Evaluate

SOL UTION: Let

so that

We find dx in the integral but not 3 dx. A 3 must be included in the integral to fulfill the requirements of du.

In words, this means the integral

needs du so that the formula may be used.

Therefore, we write

and recalling that a constant may be carried across the integral sign, we write

Notice that we needed 3 in the integral for du, and we included 3 in the integral; we then compensated for the 3 by multiplying the integral by 1/3.

Then

EXAMPLE: Evaluate

SOL UTION: Let

so that

Then

PRACTICE PROBLEMS: Evaluate the following integrals:

ANSWERS:




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