73. Two methods Evaluate ∫ dx/(x² - 1), for x 1, in two w...

Question

73. Two methods Evaluate ∫ dx/(x² - 1), for x > 1, in two ways: using partial fractions and a trigonometric substitution. Reconcile your two answers.

Accepted Answer

Welcome back, everyone. In this problem, we want to evaluate the integral of 1 divided by X2 minus 4 with respect to X for X greater than 2 using the partial fractions. Now, if we're going to evaluate this, first, let's check and factor the denominator, OK? Now we know that X2 minus 4 is a difference of 2 squares which can be rewritten as X minus 2 multiplied by X + 2. So that tells us the integrant is a rational function with distinct linear factors in the denominator. Also, the degree of the denominator. OK, is greater than the degree of the numerator, so partial fractions applies. So that means we can assume then that 1 divided by X2 minus 4 is equal to a multiplied by divided by X minus 2 plus B divided by X + 2. That means we can go ahead to solve for A and B to figure out what our fraction would decompose into. So, let's go ahead and multiply through our entire equation by X minus 2 and X + 2. When we do that, Then we should get A multiplied by X + 2, plus B multiplied by X minus 2 to be equal to 1. Now, if we expand in our terms here, OK, we're gonna have AX + 2A. Plus B X minus 2b to be equal to 1, which means that if we factor here a plus b multiplied by X. Plus 2 A minus 2B will be equal to 1. Now, if that's the case, if we compare both sides of our equation, OK, then this means that for our X terms, let me put that in red actually. For our excerpts. On the left hand side, A plus B would be equal to 0 from the right hand side because there is no X term on the right hand side. And if A + B equals 0, this means that A equals negative B or we can say that B equals negative A. On the other hand, for our constant. We can tell that 2A minus 2B on the left will be equal to 1 on the right. Now, since B equals negative A, really this is saying that 2A minus 2 multiplied by negative A. is going to be equal to 1, and that's gonna be 2A plus 2 A equals 1. So 4A equals 1. And if 4A equals 1, then A equals 1/4. And if A equals 1/4, then B would be equal to -14. So our expression. Yeah, or expression 1 divided by X2 minus 4 can be rewritten as 1/4 of 1 divided by X minus 2. -1 divided by X + 2. And no, we can use this to help us integrate because this would imply then that our integral. Of EX divided by X2 minus 4. Can be rewritten as 1/4 of the integral of 1 divided by. X minus 2 with respect to X. Minus 1/4 of the integral of 1 divided by X + 2 with respect to X. Now what are those integrals going to be? Well, for our first, it's going to be 1/4 of Ln of the absolute value of X minus 2, and for the second it's going to be 1/4 of LN of the absolute value of X + 2 all plus C. and all based on our laws of logs we can combine our logs. To become 1/4 of Len of the absolute value of X minus 2 divided by X + 2 R + C. But now remember we were told that x we can take X to be greater than 2, and since X is greater than 2, that means both X minus 2 and X + 2 are positive, so the absolute value science can be dropped, so we can rewrite or integra then as 1/4 of LNX minus 2 divided by X + 2 + C. Thus, this is going to be our integral doing it by using partial fractions. Thanks a lot for watching everyone. I hope this video helped.

73. Two methods Evaluate ∫ dx/(x² - 1), for x > 1, in two ways: using partial fractions and a trigonometric substitution. Reconcile your two answers.

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