Integral+Properties+and+Techniques

Power Rule

General Formula:

This is similar to the power rule used for differentiation, just the opposite steps.

How to use the power rule:

1. add 1 to the exponent of each degree of the polynomial 2. divide the term by your new exponent 3. add c

example

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Anti-differentiation using U - substitution

- U - substitution should be used when the derivative of an inner function is also found within the function. eg.

steps to U - Substitution 1. identify a suitable U (more on choosing U later) 2. derive U do get dU in terms of U and dx 3. isolate dx as a function of x and dU 4. cancel like terms to simplify function 5. integrate simplified function 6. sub U back into the function 7. use FTOC to reach answer (if necessary)

video on U-sub media type="youtube" key="qclrs-1rpKI" height="385" width="480"

examples

1.

choose a U that has a derivative elsewhere in the function derive U to find dU in terms of x and dx rearrange variables to get dx in terms of du and x

substitute u and dx in terms of du and x into the original expression

simplify and cancel like terms and integrate simplified term

sub in the initial U back into the function

=

In this last example choosing U was quite easy, but there are times when it becomes more difficult. You may have to try a few different things before you get what you need, but remember you will attempt to cancel the derivative with a term in the numerator of the function.

example 2.

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Sources: http://www.sitmo.com/latex/ http://wikipedia.com http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/usubdirectory/USubstitution.html

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