23-64. Integration Evaluate the following integrals.
47....

Question

23-64. Integration Evaluate the following integrals.

47. ∫ (x³ - 10x² + 27x)/(x² - 10x + 25) dx

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says determine the following interval, and we're given the interval of the quantity of X cubed minus 8X2 plus 20 X in quantity divided by the quantity of X2 minus 8 plus 16 in quantity dX. Now, the first step in determining this integral is to divide our denominator into our numerator. Since our numerator has a degree of polynomial that is actually higher than our denominator. So that means we're going to take X2 minus 8 X. Plus 16, and divided into. X cubed Minus 8 x 2. Plus 20 x and we're going to do this by long division. So x2d will go into X cubed x times. So we'll take x and multiply it by the quantity of X2 minus 8 X plus 16 in quantity, and that will give us X cubed minus 8 x 2. Plus 16 X and so we'll subtract this. From X cubed minus 8X2 plus 20 X, and this will give us 4 X as our remainder. So that means that the interval. Of the quantity of X cubed minus 8 x 2 plus 20 X in quantity divided by the quantity of X2 minus 8 X plus 16 in quantity DX is going to be equal to the integral of X DX. Plus the interval of 4 x divided by, and here we notice that our denominator in our original expression is factorable so that factors into the quantity of X minus 4 in quantity squared, and that gets multiplied by d x in our expression. So now what we want to do is to simplify that second integral using partial fractions. So we call for partial fractions that 4 X divided by the quantity of X minus 4 in quantity squared is going to be equal to. A divided by the quantity of X minus 4. In quantity plus B divided by the quantity of X minus 4 in quantity squared. So now we need to solve this for A and B, and so we need to multiply both sides of this equation by the quantity of X minus 4 and quantity squared. In doing so, we'll get that 4 X is equal to a multiplied by the quantity of X minus 4 in quantity plus B. So multiplying everything out on the right hand side, we get that 4 X is equal to AX minus 4A plus B. So now we need to match up the coefficients. On both sides of the equations for each power of X. So for the linear term, we see that A is equal to 4, and for the constant term we see that minus 4A plus b is equal to 0. So we already have one value, which is a is equal to 4, and we can substitute that into a second equation to find B. So doing that substitution, we'll have minus 4 multiplied by 4, plus B is equal to 0, and solving for B, we see that B is equal to 16. So that means that our integral, our original interval, which is the integral. Of the quantity of X cubed minus 8 x 2 plus 20 X in quantity, divided by the quantity of X2 minus 8 X plus 16 in quantity. DX is going to be equal to. Here we have the integral that we had earlier, which is going to be the integral of XDX. Then we'll have plus, and here now we have two additional integrals. So we'll have the integral of 4 divided by the quantity of X minus 4 and quantity DX. Then we'll have plus the integral of 16 divided by. The quantity of X minus 4 in quantity squared DX. So now we just need to perform the integration for each of these integrals. In doing so, this is going to be equal to. When we integrate X with respect to X for the first integral, we'll get x 2 divided by 2. Then we'll have + 4 multiplied by, and here when we integrate 1 divided by the quantity of X minus 4 in quantity, recall that we get the natural log of the absolute value of the quantity of X minus 4 in quantity, then we'll have plus 16. Uh multiplied by, and here we're integrating 1 divided by the quantity of X minus 4 in quantity squared, and recall that's going to give us -1 divided by the quantity of X minus 4 in quantity. And then we have to have an integration constant of plus C, where we have combined the integration constants from each integration into a single integration constant. That we just need to do is to simplify this, so just rewriting our expression here. This is going to be equal to 1/2 multiplied by x2 plus 4 multiplied by the natural log of the absolute value of the quantity of X minus 4 in quantity minus 16 divided by the quantity of X minus 4 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.

23-64. Integration Evaluate the following integrals.
47. ∫ (x³ - 10x² + 27x)/(x² - 10x + 25) dx

Key Concepts

Polynomial Long Division

Integration of Rational Functions

Recognizing Perfect Square Trinomials

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