Integration+by+Parts+(BC)

Or Anti-Product Rule

This method is for when you're integrating and u-sub doesn't work. This is the formula you would use:

looks more complicated than it is.

Take: this is your

so you make one part u and one part dv (choose the part that has an easier derivative for u). So for this one: and Now you need the rest of the parts for the equation; v and du. I use a 'table' for this

du is the derivative of u, and v is the integral of dv. u = 3-5x du = -5dx Then v is just the integral of dv v = sinx So now you have everything you need, so plug the pieces back into the original equation.
 * u = 3 -5x || v = ?? ||
 * du = ?? || dv= cosxdx ||
 * u = 3-5x || v = sinx ||
 * du = -5dx || dv = cosxdx ||



The rest is all very do-able. Solve for the integral than you have: This is you answer (no need to simplify, unless its multiply choice this is calculus)

[[image:kwcalculus/multil choice 1.gif width="518" height="284"]]#2[[image:kwcalculus/multi choice 2.gif width="483" height="238"]]
FREE RESPONSE

After you start this problem you may realize that you have to do it again, because the integral part of the answer needs to be solve with the integration by parts solution. So then you do it again and you see, well you have to it again. And then...Wait there's a short cut that is much easier. To start make a table, the first column just has positive and negative signs alternating starting with positive. The second column you start with the u and put its derivative below and its derivative below that until you can't derive anymore. The third column starts with dv and instead of deriving, integrate. For this application is it best to choose a u that will no longer be able to derive.

Then from left to right your answer is written out by multiplying whats in the second and third column and using the positive and negative signs to add or subtract them, you draw a lines that look like this This is your answer