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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:
