Integration by Parts Example2 Part2 So now that we have this, I’m sorry it’s a little crowded, we can simplify all of this stuff and that’s all. So this here, the negative 2x-1/2 is going to come out in front, we’re going to have -2x and let’s actually do this. So if you remember, let’s do this -2 and lnx, we’ve got all of this parts of the problem, we’ve got the lnx, we’ve got the -2, we just need the x-1/2. since the exponent her is negative, which we never want to have at our final answer, if you move it to the bottom, it’s a positive so this is actually going to be, we can change it to be -2/ x1/2 right, so we flip the sign to a positive so that’s good but then you also remember x1/2 is the same thing as the vx. So this, instead of -2/x1/2, it’s going to be -2/vx. so let’s go ahead an d write that. This became -2/vx. so this now is actually, we pull the -2 out in front, we’ve got the lnx and we’ve divide it by the vx. so that’s how we simplify this first half of the problem. Then for the integral here, we have the following, we subtract because we’ve got the minus sign there and then we can bring the -2 out in front because it’s a coefficient on dx and when you’re taking the integral of something, you’ve a coefficient like this and there’s only one term in the problem which everything is multiplied together here so it’s all considered one. You can bring the coefficient out in front, so we have minus -2 and then we can write the integral, of course minus -2 is just going to be a positive 2 right, minus a negative is a positive so let’s go ahead and write +2 times the integral. So now we have, x-1/2 and multiplied by 1/x, so that’s going to look like this x-1/2/ x, we’ve brought the -2 out in front, these are the only two thing left, the x and the -1/2 is on the top of the numerator and then the x here since it’s 1/x is in the denominator. So this is our integral. The way that we can simplify this is just so kind of complicated but the way that we can simplify this, when you have a fraction, we’ve got two like terms, x and x, we can change this and combine this into one x term by subtracting the exponents, you do when they’re in a fraction like this you do the exponent on the top minus the exponent on the bottom, we’ve got an implied one here as the exponent to put on this x. so this is actually and this becomes x and then -1/2 - 1. So -1/2 - 1 is -3/2, so we can actually go ahead and change this to be x -3/2. So that’s something we can actually take the integral of and I’m going to go ahead and erase some steps up here so that we can use this part of the board. So let’s go ahead and bring this back up here. We need to take the integral of this which we can’t, it’s finally simple enough that we can take the integral of it. So let’s go ahead and write this stuff again, we’ve got -2lnx/vx + 2—an d then let’s go ahead and draw a big parenthesis because we’re going to be multiplying the two by everything that’s comes out of the integral here, since we got it out front. So we go ahead and write dx and you remember whenever we take the integral, we add one to the exponent so -3/2 + 1 is gong to be -1/2 an d then we do the coefficient which here one is implied, the coefficient one divided by the new exponent which is -1/2. So 1 divided by -1/2 and then of course plus c which we always add whenever we take an integral no matter what. So all we have to do now is simplify. We finally take the integral and it’s just basic simplification so, this is going to be—let’s deal with this 1/-1/2 here. 1/-1/2 is going to be -2, 1/1/2 is two and we’ve got a negative sign here so that’s going to be a -2. We can change that and then this x to the -1/2, you remember we can move this x-1/2 to the bottom of a fraction and change this exponent from a negative to a positive which we’re going to wan t to do because you never want a negative exponent in the final answer, so instead of x-1/2, lets’ make this 1/x1/2 and then, x1/2 is the same thing as the