7–84. Evaluate the following integrals.
47. ∫ [(2x³ + x²...

Question

7–84. Evaluate the following integrals.

47. ∫ [(2x³ + x² - 2x - 4) / (x² - x - 2)] dx

Accepted Answer

Welcome back everyone. In this problem, we want to evaluate the integral of 3 XQ minus X2 minus 19 X minus 7, all divided by X2 minus X minus 6 with respect to X. Now, for us to evaluate this integral, since the degree of the numerator of the integran is greater than that of the denominator, then we can simplify it by performing long division. That will make it easier for us to evaluate. So, by long division, OK. By a long division. Then we're really dividing. 3 x cubed, eh. Minus X2 minus 19 X. 7 B X minus X minus 6. Now, to do that, let's start with our leading term. X2d goes into 3 X cubed 3 X times. So we put that over 3 X and then we multiply through. So that's gonna be X2 minus X minus 6 multiplied by 3 X, which gives us 3 X cubed minus 3X2 minus 18 X. OK. And now, when we subtract that, 3 X cubed minus 3X cubed is 0, negative X2 minus -3X2 is 2 X2. And -19 X minus -18X is negative X. Then we can bring down our minus 7. Next, X2 goes into 2 X 22 times, and when we multiply X2 minus X minus 6 by 2, we get 2 X2 minus 2X minus 12. No, when we subtract that polynomial, then 2 X2 minus 2X is 0, negative X minus -2 X is X, and -7 minus -12 is 5. So what does this tell us? Well, by long division, we've now found that we can rewrite our expression as 3 x + 2 plus, let me put this in blue here X + 5 divided by X squared minus X minus 6. Now, to make it simpler, all we need to do is to rewrite our term or with the remainder or rewrite our remainder in terms of partial fractions. How can we do that? Well, as a partial fraction, uh, I think let me make some space here. As a partial fraction. OK. Then, oh you know what, let me come back up here just to make some space on the page. So let me put it here and let me just put a little line here, OK? Now as a partial fraction. We could rewrite this as X + 5 divided by X minus 3 multiplied by X + 2 because X2 minus X minus 6 is equal to X minus 3 multiplied by X + 2. OK. So now, here, when we do that, then we could write this then as a divided by X minus 3 plus B divided by X + 2. If we multiply both sides by X minus 3 multiplied by X + 2, then we get X + 5. To be equal to a multiplied by X + 2 + B multiplied by X minus 3. Expand that, we get it to be A X + 2 A + BX minus 3B. I know when we combine our like terms, then X + 5 is equal to A + B multiplied by X + 2 A plus, plus 2 minus 3 B. Well, if that's the case. Then we can solve for A and B using simultaneous equations because no, we'll get A plus B to be equal to 1 by comparing our coefficients, and we'll also get 2 A minus 3B to be equal to 5. So we have two equations. And if A plus B equals 1, then that means B is equal to 1 minus A. So we could use this in our second equation to solve for A. So now when we do that. Then we're gonna get 2 A, so from equation 2. OK, from equation 2, we're gonna get 2A minus 3 multiplied by 1 minus A to be equal to 5, which is gonna be 2A minus 3 + 3A on our left-hand side, which means that 5A is going to be equal to 8. Which tells us that a then is going to be equal to 8/65. On the other hand, if A equals 8/5, then B will be equal to 1 minus 8/5, which is equal to -35. So what does that mean? We know, by decomposing into partial fractions, then we can say that x + 5 divided by X2 minus X minus 6 is equal to 8 divided by 5 multiplied by X minus 33 divided by 5 multiplied by X + 2. So now that we've decomposed our remainder into partial fractions, then we can go ahead and evaluate our integra because no, this tells us that we'll really be finding then the integral of 3 X + 2 + 8 divided by 5 multiplied by X minus 3. Minus 3 divided by 5 multiplied by x + 2 all with respect to x. So let's evaluate our integral by integrating each term. No, the integral of 3 X is going to be 3 x 2 divided by 2. The integral of 2 is 2 X. The integral of 8 divided by 5 multiplied by X minus 3 will be 8/5 of the natural log of the absolute value of X minus 3. And likewise, the integral of 3 divided by 5 multiplied by X + 2 will be 3/5 of the natural log of the absolute value of X + 2 plus some constant C. Thus this is going to be our result. This is our final answer. Thanks a lot for watching everyone. I hope this video helped.

7–84. Evaluate the following integrals.
47. ∫ [(2x³ + x² - 2x - 4) / (x² - x - 2)] dx

Key Concepts

Integration

Rational Functions

Partial Fraction Decomposition

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