23-64. Integration Evaluate the following integrals.
41....

Question

23-64. Integration Evaluate the following integrals.

41. ∫₋₁¹ x/(x + 3)² dx

Accepted Answer

Hello there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Evaluate the integral, the integral from -2 to 0 of Z divided by parentheses Z + 42 Dz using partial fraction decomposition. Awesome. So it appears for this particular problem we're asked to evaluate the specific integral using partial fraction decomposition. So that means, as the problem suggests, we will need to recall how to use partial fraction decomposition in order to solve this problem. So with that in mind, our first step that we will need to take in order to solve this problem is we will need to set up the partial fraction decomposition for Z divided by parentheses Z + 4 to 2. And this will be equal to using our partial fraction decomposition, this will be equal to a divided by Z + 4. Plus B divided by parentheses Z + 4 to the power of 2. Fantastic. Our second step now is we need to multiply both sides by parentheses Z + 4 to the power of 2 to clear our denominators, and when we do, we will find that Z is equal to a multiplied by parentheses Z + 4. Plus B, fantastic. So, Our third step now is we need to expand and collect like terms, and when we do that, we will find that Z is equal to A multiplied by Z plus 4 multiplied by A. Plus B. Fantastic. Our fourth step is we need to match coefficients for powers of Z. So for powers of Z we will find that 1 is equal to A, and for our constant, which I'll just denote as C, 0 will be equal to 4A or 4 multiplied by a plus B. Fantastic. Our step 5 is we need to solve for A and B. We will note that A is equal to 1, and we will also note that 0 is equal to 4 multiplied by 1 plus B, so when we rearrange this expression to island solve for B, we will find that B is simply equal to -4. Fantastic. So moving right along here, our step 6 is we need to rewrite the interval using the decomposition. And when we do that, we will find that the interval from -2 to 0 of Z divided by parentheses Z + 4 to the power of 2. D Z will be equal to the interval from -2 to 01 divided by Z + 4 minus 4 divided by parentheses Z + 4 to the power of 2. D Z Fantastic. Our 7th step is we need to integrate each term separately, so we will first take the integral of 1 divided by Z + 4 DZ, which will be equal to the natural log of the absolute value of Z + 4 + C subscript 1, which I'll just call C1 for short. And for our second interval, we will take the interval of 1 divided by parenthesis Z + 4 to the power of 2. D Z, which will be equal to -1 divided by Z + 4 + C2. Fantastic. So moving right along here, our 8th step is we need to combine the results, and when we do, we will be able to write the integral from -2 to 0 of Z divided by parenthesis Z + 4 to the power of 2 D Z is equal to square brackets of the natural log of the absolute value of Z + 4. Plus 4 divided by Z + 4, evaluated from -2 to 0, fantastic. So now, For our step number 9, so our 9th step, we need to evaluate the definite interval from Z equals -2 to Z equals 0. So we need to plug in the upper limit, which the upper limit, which I'll call UL or upper limit will be Z equals 0. So we're plugging in our upper limit of Z equals 0, and when we do, we will find that the natural log of the absolute value of 0 + 4 + 4 divided by 0 + 4, which will be equal to the natural log of 4 + 1. And now we need to plug in our lower limit, which I'll call LL, and our lower limit will be Z is equal to -2. And when we do, we will find that the natural log of the absolute value of -2 + 4. Plus, 4 divided by -2 + 4 will be equal to the natural log of 2 + 2. Fantastic. So now at this point, we need to subtract the lower limit result from the upper limit result, and when we do, we will achieve the following parentheses the natural log of 4 + 1 minus parentheses the natural log of 2 + 2, which will be equal to the natural log of 4 minus the natural log of 2 + 1 minus 2. And now, at this stage, we need to simplify the logarithms and fractions. So we need to take the natural log, so we're taking the natural log of 4 minus the natural log of 2, which will be equal to the natural log of 2. And 1 minus 2 will be equal to -1. So that will mean that the natural log of 4 minus the natural log of 2 + 1 minus 2 will be equal to the natural log of 2 minus 1, and that without a doubt is our final answer. Hooray, we did it. So going back to recap what we've learned in looking at the problem itself, our final answer for our for our evaluated interval will have to be once again the natural log of 2 minus 1. And that's it. That is our final answer. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

23-64. Integration Evaluate the following integrals.41. ∫₋₁¹ x/(x + 3)² dx

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23-64. Integration Evaluate the following integrals.
41. ∫₋₁¹ x/(x + 3)² dx