Evaluate the integrals in Exercises 53–76.
65. ∫3dr/√(1-...

Question

Evaluate the integrals in Exercises 53–76.

65. ∫3dr/√(1-4(r-1)²)

Accepted Answer

Welcome back everyone to another video. Evaluate the integral 4DY divided by the square root of 16 minus 36 Y minus 12. For this problem, let's remember that the integral of d x. Divided by The square root of 1 minus x 2 is inverse of sin of x plus a constant of integration c. So what we're going to do is simply use this integral and readjust the given integral in terms of this form using use substitution. To do that, let's consider the denominator. We can omit the radical for now and simply consider 16 minus. 36 y minus 12 So we want to ensure that 16 becomes 1. What we can do is simpl factor out 16. And in parentheses, we will have 1 minus 36 divided by 16. We can divide both sides by 4, right? So that'd be 9 divided by 4. In parentheses y minus 12. Additionally, we want to ensure that our variable part is a square overall, so 9 divided by 4 can be expressed as 3 halves squared, and we're going to get 16 1 minus. 3 halves y minus 1. Squared. And now let's go ahead and rewrite the integral. We have integral for D Y divided by. The square root of 16 gives us 4 in the denominator, and under the radical, we have 1 minus. We have y minus 1. Squared Notice that we're gonna cancel out. The constants which are equal to 4, and this is where we can introduce the u substitution. Let's suppose that u is equal to. 3 halves y minus 1 and then we can show that the derivative of u with respect toy is going to be equal to 3 halves. So now, what is D Y? Well, we can use cross multiplication and show that the Y is equal to DU multiplied by 2 divided by 3. So 2/3. The EU. We can now rewrite the integral. Dy becomes DU multiplied by 2/3. So let's go ahead and factor out 2/3. And in the denominator. We're going to have the square root of 1 minus u2. Perfect. So now we have that elementary integral. Its value is 2/3 inverse of sin of u plus c, which is 2/3 Inverse of sin of. Now we know that u is 3 multiplied by y minus 1 divided by 2. So we substitute and we're going to add the constant of integration c. And this is how we get our final answer for this problem. Thank you for watching.

Evaluate the integrals in Exercises 53–76.65. ∫3dr/√(1-4(r-1)²)

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Evaluate the integrals in Exercises 53–76.
65. ∫3dr/√(1-4(r-1)²)