This is a video from everystepcalculus.com demonstrating how my programs work on a ti 89 titanium calculator and other calculators in the t i system for physics and calculus problems integration by parts and with regard to cosine such as the Integral of x^2*cos(3x) and ahh. Let’s get started to get to my menu you have to push 2nd alpha and put the letters i n d e x
in here and push alpha to put the eight and the open and closed parenthesis press enter and you’re into my menu you scroll down to what your interest was what you need and we want integration by parts here so were going to click on that and generally you go to integration by parts when you’re doing any transcendentals like log of x or sine of x cosine of x, e to the x and we click on that right now we want to do a cosine so I give you that choice and we’re going to do this example right here x squared times cosine of three x which takes two integration by parts calculations to get the answer and I’ll take you through that you have to press alpha before you enter anything into my menus here got to press alpha x squared times cosine of three times x. I always show you what you’ve entered in case you’ve made a mistake you can check it and do it over say it’s ok here’s the integral we’re trying to find here you have to always do this procedure for integration by parts you have to find d v they give you dv but choose generally the hardest integral in here which is cosine of three x and then do the integral of that which is sine of three x divided by three and then you find u which is the other part here is x squared you find u and then do the derivative of that which is two x d x and here’s the formula here’s what we’re after finding the integral of this but the formula is v times u minus the integral of v times d u dx so we start puttin the parts together sine of three x divided by three the v part and then u part which is the x part minus the integral of v which is sine of three x divided by three times two x again you’ll notice that when ever you have this like for instance dividing by three you have to bring this outside the integral here before you can integrate something here we have a two also so you’re going to bring that outside the integral first and that’s where I’ve gotten the two thirds out here and then you got this integral here remaining but you need to do integration by parts again because you have a times sign in here you can’t integrate when you have a times sign or divide sign you have to change it somehow or some other formula ecetera so we need a second integration by parts we choose the sign of three x here for dv integral of that is minus cosine three x divided by three you write all this stuff on your paper exactly as you’re doing it if you doing a test or homework and you get a hundred percent on the problem and here’s u right here which we’re going to find the derivative of which is one and we’re gonna to use this integral and do the formula again v times u minus the integral of v times du so we put it together and we put two thirds you notice that we have to bring the one third out of the integral again and so we do that here. I show you how to do that and then you have the first part of the formula and the second part which is multiplied two ninths time the minus cosine of that and we do it again to change things to do the calculations of it and then we have to bring the one third out again after we do the integral we’re doing the integral of minus cosine which turns out to be minus sine of three x divide by three pretty complicated. I don’t know how you’d do this without my program but I guess it could be on a test and so then therefore you’re going to find the answer here which is equal to x squared you bring the first part remember we did a first part and then did the second integral or second integration by parts so here’s the first part and then we add the second integration here and then multiply it out and here’s the answer here. Pretty neat, huh? everystepcalculus.com. Go to my site buy my programs and pass calculus.