Definite integrals Use a change of variables or Table 5.6 to e...

Question

Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.

∫₂/₍₅√₃₎^²/⁵ d𝓍/ x√(25𝓍²― 1)

Accepted Answer

Hello. In this video we are going to be determining the definite integral. So we are given the integral of the boundary 3 square root of 2 divided by 8 to the larger boundary 32 root of 2 divided by 2. Of the integrand D Y divided by Y multiplied by the square root of 16 y2 minus 1. Now, we're going to approach this problem by using a trigonometric substitution. But since we are making a substitution, it's going to be a bit difficult to work with the current bounds. Well, recall that the normal version, or the normal way we approach a trigonometric substitution, is that we have a function in terms of Y and transform it into a function of theta. So we're going to need to transform the boundaries from respect to Y to respect of theta. But at the end of the problem, we want to go from terms of theta back into terms of Y. So, that means that whatever we start with, we are going to end up with the same boundaries at the end of our results. So for now, let's go ahead and focus on solving the given integral. Without the boundaries. So, we have the integral of DY, which can be written as 1 D Y. Divided by Y multiplied by the square root of 16 Y2 minus 1. Now, if we take a look at the square root, the square root is in the form of X2 minus A2, and that means our trigonometric substitution is going to be of the form X is equal to a sequent theta. Now, here, our X2 term is equal to 16 Y 2, and if we take the square root of both sides of this equation, we get X is equal to 4 Y. So, our X term for our trigonometric substitution is going to be for Y. Now, in this case here, our A square term is just going to be one, so we have a coefficient of one for our trigonometric substitution. So, by applying these terms, our trigonometric substitution is going to be defined as 4 Y equal to sequent of theta. Now, in this case here, we need a substitution for only Y. So if we divide both sides of this equation by 4, we get Y is equal to 1/4 sequent theta. Now what we need to do is we need to come up with a substitution from of Dy into terms of theta. So if we take the derivative of both sides of this equation with respect to Y, we are left with D Y is equal to 1/4 sequent theta tangent theta D theta. And so now what we are going to do is we are going to apply our trigonometric substitution to our integral. So by applying these substitutions, we can rewrite the integral to be the integral of 14th sequent theta. Tangent data. The Theta Divided by the quantity y which is 14th. Sequent of theta multiplied by the square root of 16 multiplied by y2 which is going to be 1/16 squad theta minus 1. Here we can do a few a few simplifications. Notice that we have a 1/4 see theta in both the numerator and denominator. We can go ahead and eliminate these terms to be 1. Now, in the square root, we have 16 multiplied by a coefficient of 1 divided by 16. That is going to simplify to be 1, leaving us with a denominator of the square root of sequent squared theta minus 1. And by using the Pythagorean identity, we can rewrite this to be the square root of tangent squared theta, and the square root of tangent squared theta is going to simplify to be tangent of theta. What this means is that our integral is going to simplify to be the integral of tangent of theta, divided by tangent of theta D theta, and this is going to further simplify to just be the integral of 1 D theta. Now, if we were to take the integral of 1D theta, that is going to give us an antiderivative of just theta, and this is going to be evaluated with some bounds from B to A to B, because we were working with a definite integral. But now this is going to be the step where we want to rewrite this term of theta back into terms of Y. Now recall that the original trigonometric substitution we were working with was 4 Y is equal to see of theta. We need to solve for theta back in the terms of Y, and in order to do this, we are going to take the equin inverse of both sides of our trigonometric substitution. That is going to leave us with theta is equal to see inverse of 4 Y. And so now if we plug this back in for theta, we can rewrite our antiderivative to be sequent inverse of 4 Y. And since this is the antiderivative with respect to Y, we can now replace EMB with our original bounds. Remember that our smaller boundary is going to be 3, square root 2 divided by 8, and our upper boundary is going to be 322 divided by 4. Now, all we need to do is we just need to evaluate this quantity. So by plugging in the boundaries, the value of the integral is going to be sequent inverse of 4 multiplied by 3 square root 2 divided by 4, or divided by 2. I put the wrong denominator here. Minus sequent inverse of 4 multiplied by the smaller boundary, which is 3 square root 2 divided by 8. And by simplifying these terms, this will simplify to be sequent inverse. Of 6 square root of 2 minus sequent inverse of 3 square root of 2 divided by 2. This is going to be the simplest form of the solution to the given definite integral, and this is going to be the answer to the problem. And with that being said, the overall solution to this problem is going to be C. So I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.

∫₂/₍₅√₃₎^²/⁵ d𝓍/ x√(25𝓍²― 1)

Key Concepts

Definite Integrals

Change of Variables

Integration Techniques

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