 In this video, we're going to discuss the solution of question four from the midterm practice exam for calculus to math 1220. This one we're asked to find which substitution should be used in the calculation of this integral right here. Well, investigate what type of substitution are we talking about. There's a couple substitutions we've talked about this semester. U substitutions, trigonometric substitutions, and rationalizing substitutions. Well, because of the square root of x squared minus 25 on top, this to me indicates that we should be using a trigonometric substitution. I see that it's a difference of squares where the variable x squared comes first and then the constant 25 shows up second. So since it's a function squared minus a constant square, that to me indicates a secant substitution. So I'm not going to use sine. I'm not going to use tangent. And so I've already eliminated down to choices C and D. So which one do we use to do x equals five secant theta or do we use x equals one fifth secant theta? Well, the difference there is that when you're looking at this difference of squares, you have to take the square root of the function squared. You have to take the square root of the constant squared right there. And so the left-hand side, when you take its square root, you're going to end up with an x. The right-hand side, when you take its square root, that'll go on the right, in which case you then get the coefficient of secant. So this is actually going to give us choice C as our answer. Now, what variation would have actually given us choice D as the final answer? What we could have seen instead would be the following. If we were integrating the function, the square root of 25x squared minus one over x dx, in this situation, we'd actually would have gravitated towards x equals one fifth secant. Because notice it's still a difference of squares where the variable comes first. That still gives us a secant substitution. But when you take the square root of the first one, that gives you a five x, the square root of the second one is a one. And so that actually would have given you a five x equal to secant theta, which if you solve for x, you'll get this substitution right there. So that gives you a situation where D would actually be the correct answer. But as the question was stated, C is our correct answer.