Integrals Example Number 9 The new problem is another two problem, 2x* vx -1/vx dx. First thing I want to do is convert this to something that’s easier to integrate. As a rule we say that in the previous video the vx of is the same thing as x1/2 always. So we are going to change this to 2x (x1/2- 1). Let me explain this for a second x-1/2. So normally we would have x at 1/x1/2 but by converting this text to the ½ but instead since we don’t want to do with fractions I want to head it in the same step moved it to the top and flip the exponent which we also explained in previous videos. So instead of 1/vx this becomes 1* x-1/2.And then of course dx. So now, we simplify this before we integrate when you are multiplying x together like this instead of multiplying the exponents you add them, this is a 1 implied there for the exponent. So we are going to add those so it’s going to be the integral of 2x 1+1/2 is 1-1/2 or 3/2 so this is going to be x3/2 minus, of course this one is implied here so we can drop that and we’re just going to say x-1/2 dx. Now this is something going actually integrate. So we’re going to go ahead and take it term by term. We’re going to write our x down here and we’re going to add 1 to the exponent 3/2+ 1. We’re going to say + 2/2 which is the same as 1 adding 1 3+2= 5/2 so 5/2 is the exponent. And then we’re going to take 2 and we’re going to divide by 2 being the coefficient and we’re going to divide by the new exponent 5/2, 2 over 5/2 is the same as 2*2/5 which is 4/5. So we are going to going to go ahead and make this 4/5. Let’s simplify that -x -1/2+1 is ½ and then that could be coefficient divided by the new exponent. The coefficient here implied is 1 so it’s 1 over 1/2, 1 over 1/2 is the same as 1*2/1. 1*12/1 is 2 so we will change that coefficient. Simplify it to be 2 and then of course as always +c for the constant and this is the final answer.