ln(2x) Integration by Parts

This video is gonna be on integration with regards to natural log. It is very difficult to me, maybe no to you, but to me it’s very difficult. You never use it so it’s tough to come up with a test or homework. So it’s good to have a program like this. That’s why I programmed it myself. So let’s get started, 2nd alpha to enter formula to get to the menu, this tells you that you can enter letters in the titanium calculator. So we enter index 8 () and press enter. Scroll to integration by parts or if you wanted to find out about log of x. So I ask you in log of x, what do want to do? Differentiate, integrate or solve for x. Some tests have solving for x and log of x which is another tricky thing if you’re not used to it. That will be in another video. So we want to integrate, so we select integrate. We are in integration by parts, when you come to log of x and integrating we will do log or x which is In(x), select and put in the example. You should see: Evaluate a In(x) Integral Exmp: x^2* In(x) Remember to press alpha to enter the formula: x^2*In(x) ,then press enter. It shows what you are trying to integrate which should be: ∫[x^2*In(x)]dx, then select ok The function will appear as follows: ∫[x^2*In(x)]dx dv = (x^2)dx v = ∫[x^2]dx = x^3/3 u = in(x) du = (1/x)dx Notice I chose the dv which is already the derivative and v would be the integration of that. So dv is gonna be x^2. And v is the integral of x^2 which is gonna be x^3/3. For those of you who don’t know how that happened, I can’t do everything exactly step by step because we’re doing function of derivatives. In the integral of the x^2, you add 1 to the 2 (the exponent) and divide by 3, so we have x^3/3. When you divide by 3 you have to multiply times whatever is in front of the x and that’s 1 so it will be 1/3 times 1/3 or x^3/3. The u choice is going to be In(x), so the derivative of In9x) is (1/x)dx. Here is the function and the formula for integration by parts ∫[x^2*In(x)]dx = v*u – ∫[v*(du)]dx =x^3/3 [In(x)] • ∫[(x^3/3) (1/x)]dx = x^3*In(x)/3 • ∫[(x^2/3)]dx Further you can see =x^3*In9x)/3 • (1/3) ∫[(x^2)]dx =x^3*In(x)/3 • (1/3) (x^(2+1) / (2+1) = x^3 In(x)/3 =-(1/3) (x^3/3) We put the 1/3 in front of the integral sign and then integrate the inside. When you’re going to integrate x^2 notice I show you we’re going to do 2+1 add 1 to the exponent and divide by 2+1. So that’s exponent 3 divided by 3. The you have your answer for function ∫[x^2* In(x)]dx Answer = x^3*In(x)/3 -x^3/9+c Check my videos out on my website, eveystepcalculus.com and of course other videos that help you. You can purchase these programs and get through calculus very easily.