Let f(x) = (4x³ + x² + 4x + 2) / (x² + 1). Use long division t...

Question

Let f(x) = (4x³ + x² + 4x + 2) / (x² + 1). Use long division to show that f(x) = 4x + 1 + 1 / (x² + 1) and use this result to evaluate ∫f(x) dx.

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says let G of X be equal to the quantity of 3 XQ plus 2 X2 + 12 x plus 9 in quantity, divided by the quantity of X2 + 4 in quantity. Use polynomial long division to express G of X in the form of A X plus B plus C divided by the quantity of X2 + 4, and then evaluate the integral of G of X with respect to X. So the first thing we need to do is to use polynomial long division to express G of X in the form of A X plus B plus C divided by the quantity of X2 + 4. So we're gonna take X2 + 4, and divide it into our polynomial, which is 3 X cubed plus 2 X2. Plus 12 X. Plus 9. The first step is to determine how many times X2 will go into 3 X cued, and that will go 3 x times. So we'll take 3 X and multiply it by the quantity of X2 + 4, and this will give us 3 X cubed. Plus 12 X, and we're subtract that quantity from our polynomial. And in doing so, we will get 2 X squared plus 9. Now we just need to determine how many times X2 will go into 2 X2. Now we'll go 2 times, so we'll add 2 to our quotient, and then we'll take 2 and multiply it by the quantity of X2 + 4. That will give us 2 X2. Plus 8 And we're subtract this quantity from 2 X2 + 9, and this will give us a remainder of 1. So that means that G of X can be written as being equal to our quotient, which is 3 X plus 2. Plus our remainder of 1 divided by the quantity of X2 + 4. So this is the answer that we're looking for for the first part of our problem. Now, for the second part, we need to evaluate the interval of G of X with respect to X. And this is going to be equal to the integral of the quantity of 3 X plus 2. Plus 1 divided by the quantity of X2 + 4 in quantity, and then that polynomials can be integrated with respect to X. So in order to evaluate this integral, we just need to integrate each of those terms in that polynomial and add them together. That means this interval is going to be equal to for the first term. We're going to be integrating 3 X with respect to X, and that will give us 3 X squared, divided by 2. So we'll have plus for the second term, we're going to be integrating to with respect to X, which will give me 2 X. Then we'll have plus the last term we're gonna be integrating 1 divided by the quantity of X2 + 4 in quantity with respect to X. Now, in order to do this integration, recall that the integral of 1 divided by the quantity of X2 plus A2 and integrating that fraction with respect to X, is going to be equal to. One divided by A, multiplied by the inverse tangent. Of the quantity of X divided by A. Now, in our case here, A is going to be equal to 2, thence 4 is equal to 2 squad. That means that when we integrate that last term, we get 1 divided by 2, multiplied by the inverse tangent. Of quantity of X divided by 2 in quantity. And then, we still need to add in our integration contents, we'll have a plus C at the end. So that is the answer that we're looking for for the second part of this problem. So just to recap, we found that G of X is equal to 3 X plus 2 + 1 divided by the quantity of X2 + 4 in quantity, and we found that the integral of G of X with respect to X is equal to. 3 x 2 divided by 2 plus 2 X plus 1 divided by 2 multiplied by the inverse tangent of the quantity of X divided by 2 in quantity plus C. So those are the answers 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.

Let f(x) = (4x³ + x² + 4x + 2) / (x² + 1). Use long division to show that f(x) = 4x + 1 + 1 / (x² + 1) and use this result to evaluate ∫f(x) dx.

Key Concepts

Polynomial Long Division

Integration of Rational Functions

Fundamental Theorem of Calculus

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