Trig substitution integrals9/20/2023 ![]() Then you can write down DX is equal to 3 cosign theta D theta, just move the D theta over there. DX, D theta, the derivative of this with respect to theta is simply, 3 times the cosign of theta, just take the derivative. You may not know exactly where I'm going with this yet. It tells you right here, the substitution is to make X is equal to A sign theta, so what you need to do is write X is equal to A, that A is 3 in this case times the sign of theta. A is equal to 3, so your A squared minus X squared, so A is equal 3. Notice that this interval kind of takes the form of this. It doesn't have to be just as expression in the interval, just something in the interval has to have this in it and then you'll be good to go. This is obviously over DX and notice that you can write this interval as the square root of 3 squared minus X squared over X squared DX. Well, like I said, you need to be on the lookout. What if you had the interval of the square root of 9 minus X squared over X squared. I'm going to draw a line around here and we're going to go and try to work a few problems, just to get a feel for how you could use this. This is the same actual identity, to solve for different thing. For this problem, it's going to be secant squared minus 1 is equal to tangent squared. These are going to be useful for these problem. The identity here is 1 plus tangent squared is equal to secant squared. This is just simply cosign squared plus sign squared 1. ![]() The identity, if you're in this case, is going to be 1 minus sin squared is equal to cosign squared. Over here, there's going to be some relevant identities that you're going to need to use for each case. ![]() Finally, if you see X squared minus A squared, then you need to substitute in X is equal is A times the secant of theta. If you see the expression A squared plus X squared, then you'll to substitute in A times tangent theta. If you see this, then you need to substitute in X is equal to A times the sign of theta. Be on the lookout for that, if you see that, you're going to need to use integration by trig substitution. What I'm going to have here is expression, that you need to be on the lookout for, the first one is the square root of A squared minus X squared, all under the square root. These problems can get quite long, but I'm going to give you a couple to give you a feel for what you're going to need to do and the general way in which you need to go in and solve them.įor integration by trig substitution, what you need to be on the lookout for are a couple of expressions, really. I'm going to write a couple of things on the board that you're going to need to know to do these types of problems and I'm going to encourage you to work a lot more problems then I'm going to work here. The section of the course we're going to work on a couple problems using Integration by Trig Substitution. View the full course and learn by working problems step-by-step! This is just a few minutes from a multi-hour course.
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