Integration by Parts Example 8

Hi everyone. Welcome back to integralcalc.com. We’re going to be doing another integration by parts problem today. This one is the integral of x to the 7th times ln or the natural log of x dx. So we're going to be using the integration by parts formula which if you remember is uv minus the integral of vdu. And what we do is assign u and dv to each term in our problem here. So the first thing we need to do is pick which one of the components of our integral will be u and which one will be dv. For integration by parts, try to pick whichever one for u will simplify itself when we take the derivative, which in this case is a pretty natural give away because ln or the natural log of x when we take the derivative of that, we get 1/x, which is much simpler than natural log of x so ln of x here ends up being a good candidate to assign u to. So we'll say u equals natural log of x and then we'll take the derivative of that and get 1/x. Because we've assigned 1/x to u, that means that dv must be x to the 7th, so we'll say dv equals x to the 7th and then v, we add 1 to the exponent because here we're taking the integral to get v. We add 1 to the exponent to get 8 and then we divide the coefficient which is an implied 1 by the new exponent, which is 8. Now that we've calculated u, du, v and dv, we can go ahead and plug them into our integration by parts equation here. So you can see, we have u times v, so let's go ahead and write this out here. We have ln of x times v, which is 1/8 x to the 8 minus the integral of vdu so we have 1/8x to the 8 times du which is 1/x and then we have to of course add dx here because it accompanies the integral as part of the integral notation. So now that we've done that, we can go ahead and simplify somewhat so I'm going to bring 1/8 out in front here as the coefficient. I’m also going to bring out x to the 8 because that should go in front of the natural log of x. So we'll have 1/8 x to the 8 ln of x minus and then here for the integral, we can bring 1/8 out in front of the integral because it's a coefficient and everything inside is multiplied together. So we'll bring it out in front and we're left with the integral of x to the 8 over x. Because we have eight xs in the numerator and one in the denominator, that will simplify to x to the 7th dx. Now this is something that we can easily take the integral of. We’ll end up with 1/8 x to the 8 ln of x minus 1/8 and then we'll draw a big parenthesis to denote that this is clearly where we're starting to write down where we took this integral. So again, with x to the 7, we add one to the exponent and then divide the coefficient, 1 by the new exponent which of course is going to be 8, 1/8 x to the 8 and then we add c to account for the constant. And now what we have to do is just simplify. What I’m going to do is go ahead and factor out 1/8 x to the 8 because I think that will be cleanest answer.  So I’ll factor out 1/8 x to the 8 and then I will multiply that by ln of x, the first term here and then all we have left is just 1/8 so then it will just be minus 1/8 plus c. And that is our final answer. I hope that helped. I'll see you guys next time. Bye.