7–84. Evaluate the following integrals.
22. ∫ [1 / ((x -...

Question

7–84. Evaluate the following integrals.

22. ∫ [1 / ((x - a)(x - b))] dx, where a ≠ b

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says to evaluate the interval of 1 divided by the quantity of X minus 3 in quantity multiplied by the quantity of X + 2 in quantity DX. So, in order to evaluate this interval, we're actually going to use partial fractions. So we want to break up our integrand into two different fractions. So we'll have one divided by The quantity of X minus 3 in quantity multiplied by the quantity of X plus 2 in quantity, and what's at that equal to A. Divided by the quantity of X minus 3 in quantity, plus B divided by the quantity of X plus 2 in quantity. And so, we're going to multiply both sides of this equation by the quantity of X minus 3 in quantity, multiplied by the quantity of X plus 2 in quantity. In doing so, we will have that 1 is equal to a multiplied by the quantity of X plus 2 in quantity, plus B multiplied by the quantity of X minus 3 in quantity. So now we need to find values for A and B. And one way to do this is, since we just have two variables here, is to set X so that the value multiplying either A or B is going to be equal to 0. So first one, we're going to set X equal to 3, so that the quantity multiplying B is going to be equal to 0. In doing so, we will have that 1 is equal to a multiplied by the quantity of 3 + 2. And quantity, plus B multiplied by the quantity of 3 minus 3. So simplifying this expression, we'll have that 1 is equal to 5 multiplied by A, and solving for A, we get that A is equal to 1 divided by 5. The other X value that we can choose is X equal to -2, so this will make the quantity multiplying our A coefficient equal to 0. And so doing this, we'll have 1 is equal to a multiplied by the quantity of -2 + 2 in quantity, plus B multiplied by the quantity of -2 minus 3 in quantity. To simplifying this this expression, we get 1 is equal to minus 5 multiplied by B and solving for B, we get that B is equal to minus 1 divided by 5. So that means that our interval of 1 divided by the quantity of X minus 3 in quantity, multiplied by the quantity of X plus 2 in quantity DX is going to be equal to the integral of 1 divided by 5. Multiplied by the quantity of one divided by. The quantity of X minus 3. In quantity, DX. Minus the integral of 1 divided by 5, multiplied by the quantity of 1 divided by the quantity of X plus 2 in quantity TX. So here we have two integrals to actually evaluate, but these are straightforward integrals that will give us natural log functions. So evaluating our integral integrals, this is going to be equal to. Our constant can be pulled outside the integral. We will have 1 divided by 5, and that gets multiplied by the integral of 1 divided by the quantity of X minus 3 in quantity DX, which will give us the natural log of the absolute value of the quantity of X minus 3 in quantity. Then we'll have minus. Again, we'll pull out our coefficient out front front of the integral, so we'll have 1 divided by 5, and this gets multiplied by the integral of 1 divided by the quantity of X plus 2 in quantity DX. And again, when we evaluate this interval, we'll get the natural log functions, so we'll have the natural log of the absolute value of X + 2 in quantity. And then we'll have to add an integration constant of plus C. So here we've combined the integration constants from both integrals into a single integration constant. Now, our last step is to combine our two natural log terms, since both are being multiplied by 1 divided by 5, we can actually combine them. So this is going to be equal to 1 divided by 5, multiplied by the natural log of the absolute value of the quantity of X minus 3 in quantity, divided by the quantity of X plus 2 in quantity plus C. So this here is the answer that we're looking for for this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

7–84. Evaluate the following integrals.
22. ∫ [1 / ((x - a)(x - b))] dx, where a ≠ b

Key Concepts

Integration

Partial Fraction Decomposition

Definite vs. Indefinite Integrals

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