?sin^m xcos^n xdx There are three rules governing how we deal with integrals of

Question

?sin^m xcos^n xdx


There are three rules governing how we deal with integrals of this form.

1.If the power of the sine is odd and positive, save one sine factor and
convert the remaining factors to cosine. Then, expand and integrate.

2.If the power of cosine is odd and positive, save one cosine factor and
convert the rest to sine. Then, expand and integrate.

3.If the powers of both the sine and cosine are even and nonnegative, make repeated use of the identities



1. Integrate ????cos?^3 (x) ?sin?^3 (x)dx? . Which of the rules above will you choose? Why?




2. Choose a rule and rewrite the above integral using it. Be sure to put the factor you are saving to the immediate left of the dx. Please underline the saved factor.



3. Convert the non-underlined terms in the result of part two into sin or cos depending on the rule you chose to use. What identity can we use to do this?





4. What relationships between sin and cos are we taking advantage of when we use the rules above? Based on step 2 above what should we set u equal to so that we can make a substitution that allows us to use the general power rule to complete the integration?

u =
du =

Explanation / Answer

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