7–64. Integration review Evaluate the following integrals.
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Question

7–64. Integration review Evaluate the following integrals.

32. ∫ from 0 to 2 of x / (x² + 4x + 8) dx

Accepted Answer

Hello. In this video, we are going to be calculating the definite integral. We are given the integral from 0 to 1 of X divided by the quantity X2 + 2 x + 5. Before we start evaluating the integral, let's go ahead and simplify the denominator. So again, the denominator is X2 + 2 x + 5. What we can go ahead and do is we can try to factor this this denominator, but currently the denominator is not in a factorable form. We can, however, rewrite this as a factorable form. See, we know that if we have X2 + 2 X + 1, this will factor to be X + 1 quantity squared, but that means that we will need to rewrite this 5 as 1 + 4. Then we will group up the first three terms together. So, if we were to factor X2 + 2 X + 1, we will now have X + 1. Quantity squared + 4, and that will allow us to rewrite this integral to be the integral from 0 to 1 of X divided by the quantity X + 1 squad + 4. Here, what we can go ahead and do is we can use a U substitution to simplify the square term. If we allow U to equal to X plus 1, then D U will equal to DX. This will allow us to rewrite. The following integral, as the integral of a substitution of X in terms of U, which we currently don't have, but we can solve for. If we solve for X in our U substitution, we get X is equal to U minus 1. So, we end up with U minus 1 divided by U2 + 4. This is going to be the substitutive form of the integral, but now we need to go ahead and rewrite our original bounds with respect to you. In order to do this, we will plug in each of the values of the bounds into our use substitution, and that is going to give us a lower bound of 1 and an upper bound of 2. So we have the integral from 1 to 2 of U minus 1 divided by U2 + 4. Now, we can further simplify this integral. By rewriting this as a difference of integrals. If we were to split up the integrand, we can rewrite the integrand to be the integral of 1 to 2 of you divided by U2 + 4 minus 1 divided by U2 + 4. Then we can go ahead and split this up into a difference. Of definite integrals. OK. Now let's go ahead and solve the first integral. Now, in order to solve the first integral, let's go ahead and use another substitution in terms of W. We will allow W to equal to U 2 + 4. Then if we were to rewrite this in terms of W by taking the derivative with respect to W, DW is equal to 2 U. To you to you. Then, if we were to move the constant term to the left hand side of this DW statement, we have one half DW is equal to UDU. If we apply this to the first integral, then the first integral can be rewritten as the integral of 1 divided. By WDW and again we will go ahead and rewrite our bounce. Oh, forgot about the constant. 1/2, so there should be a 1/2 in front of the integral. Now, we must rewrite our bounds with respect to W. So we are going to take our bounds from 1 to 2 of the first integral and plug them into our W statement. That is going to give us a lower bound of 5, and an upper bound of 8. No The integral of 1 divided by W is going to be the natural logarithm of W, so the antiderivative of this integral is going to be our constant 1/2 multiplied by the natural logarithm of W, and this is going to be evaluated from 5 to 8. By expanding this, we will end up with 1/2 multiplied by the natural logarithm of 8 minus 1/2 multiplied by the natural logarithm of 5. And this is going to further simplify to give us 1/2 times the natural logarithm of 8 divided by 5. This is going to be the value of the first integral. Now let's go ahead and evaluate the second integral. Now, the second integral is the integral from 1 to 2 of 1 divided by U2 + 4. Now, in order to evaluate this integral, we can use the identity for the tangent inverse integral. See, that integral is the following. If we have an integral in the form of one divided by U squared plus Aquad, where A is a squared constant, then this is going to be equivalent to 1 divided by A. Multiplied by tangent inverse of U divided by A plus C. In this case here, our squared constant is 4, so we have a squared is equal to 4, which means that A is equal to 2. And if we were to use this following identity, the the antiderivative of this definite integral is going to be 1 divided by A, which is 1/2 multiplied by tangent inverse of U divided by A, which is U divided by 2. And this is going to be evaluated from 1 to 2. By evaluating this, we end up with one half multiplied by tangent inverse of 2 divided by 2. Minus 1/2 tangent inverse of 1 divided by 1. And this is going to further simplify, to give us pi divided by 8 minus 12 tangent inverse of 1/2. So this is going to be the 2nd value for our 2nd integral. So, just to recap, we just solved that the second integral is pi divided by 8 minus 1/2 multiplied by tangent inverse of a half. And earlier, the first integral was 1/2 multiplied by the natural logarithm of 8 divided by 5. If we were to plug this back into the original statement we were working with, which was the difference of the two integrals, we can rewrite this statement to be 1/2. The natural logarithm of 8 divided by 5 minus the quantity pi divided by 8 minus 1/2 multiplied by tangent inverse of 12. And this is going to further simplify by just distributing the negative sign across the quantity, and the simplest form of this answer is going to be 1/2 multiplied by the natural logarithm of 8 divided by 5, plus 1/2 multiplied by the tangent inverse of 1/2. Minus pi divided by 8, and this is going to be the simplest form of the value of the given definite integral. And with that being said, the overall solution to this problem is going to be B. 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.

7–64. Integration review Evaluate the following integrals.
32. ∫ from 0 to 2 of x / (x² + 4x + 8) dx

Key Concepts

Definite Integrals

Integration Techniques

Polynomial Division and Completing the Square

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