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

Question

23-64. Integration Evaluate the following integrals.

23. ∫ [3 / ((x - 1)(x + 2))] dx

Accepted Answer

Welcome back. In this problem, we want to evaluate the integral of 5 divided by Y + 3 multiplied by Y minus 2 with respect to Y. Now, to help us evaluate the integral, we can express our expression, I guess you could say that, as partial fractions. In other words, we could let 5 divided by Y + 3 multiplied by Y minus 2 equal a divided by Y + 3. Plus B divided by Y minus 2. So now if we can figure out what A and B are, we can rewrite our expression in terms of partial fractions and evaluate each one. So let's do that. First, let's start by setting up the equation. If we multiply through our entire equation by Y + 3 multiplied by Y minus 2. Then we should get 5 to be equal to a multiplied by Y minus 2, plus B multiplied by Y + 3. If we expand our brackets, that means 5 E equals A Y minus 2A plus BY + 3B. And if we group like terms, then 5 will be equal to A plus B multiplied by Y plus 3 B minus 2A. From this we can set up a system of equations because now if we compare both sides of our equation then for our coefficient of y this must mean that a plus B equals 0 because there is no Y term on the right hand side, so this is equation 1. And for our constant, then 3 b minus 2A must be equal to 5. Now, if we multiply equation 1 by 2 and add equation 3, OK, then we should get 2 multiplied by A plus B. All right, let me write down what I'm gonna do here. So If we multiply equation one. OK, by 2 and add equation 3. I mean, read that out clearly that this is the equation that we're working with. Then We should find that we get 2 multiplied by A + B. Plus 3 B minus 2A to be equal to 0 + 5. In that case, 5B would be equal to 5 because 2A minus 2A equals 0 and thus B equals 1. And if B equals 1, OK. OK, let me This would imply that from A + B equals 0, then A + 1 would be equal to 0, which means A equals -1. So if I make some space over here on the on the right, OK. Then if we can rewrite our integral, OK. Then the integral of 5 divided by Y + 3 multiplied by Y minus 2 with respect to Y is going to be equal to the integral of -1 divided by Y + 3 with respect to Y. Plus the integral of 1 divided by y minus 2 with respect toy. Now we can integrate each term. Now the integral of -1 divided by Y + 3 with respect to Y equals the negative LN of the absolute value of Y + 3. On the other hand, the integral of 1 divided by y minus 2 with respect toy would be equal to the log, the natural log, sorry, of the absolute value of Y minus 2. Thus, here this would be our result and if we simplify this using the property of logs, OK, or properties of logs, then we could rewrite this as the natural log of the absolute value of Y minus 2 divided by Y + 3 + C. This would be our integral. Thanks a lot for watching everyone. I hope this video helped.

23-64. Integration Evaluate the following integrals.23. ∫ [3 / ((x - 1)(x + 2))] dx

Watch next

23-64. Integration Evaluate the following integrals.
23. ∫ [3 / ((x - 1)(x + 2))] dx