2–74. Integration techniques Use the methods introduced in Sec...

Question

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

10. ∫ (x³ + 3x² + 1)/(x³ + 1) dx

Accepted Answer

Hello. In this video, we are going to be computing the following indefinite integral using appropriate algebraic techniques. Now, in this integral, we are given the integral of X cubed plus 4 X2 + 2 divided by the quantity X cubed plus 2DX. Now, when we are looking at this integral, the integran is in the form of a rational function. But notice that the overall power of the numerator matches the overall power of the denominator. So since this is a rational function where the numerator and denominator have the same powers, what we can do is we can go ahead and simplify the integrand by using polynomial long division. So, if we set up the polynomial long division, we will have the denominator, X cub plus 2, dividing the numerator, which is X cubed plus 4 X2 + 2. Now performing this polynomial long division, we have to ask ourselves, what is the quantity that we can multiply to XQ + 2 to try to match this first term? Well, in this case, we can only multiply XQ + 2 by 1. Multiplying that will give us X cubed plus 2, but then we need to subtract this quantity afterward. Subtracting this quantity is going to allow us to eliminate the X cube terms in the quotient and eliminate the constant term, and we will be left with the remainder of positive for X squared. And so by polynomial long division, we are able to simplify the integran to be the integral of 1 + 4 X squared divided by the original denominator, which is XQ plus 2DX. Here, we can go ahead and split up the integran into a sum of two integrals. So we will have the integral of 1 plus the integral of 4 X squared, divided by X cubed plus 2DX. We will go ahead and label the first integral 1 and the second integral 2. So let's go ahead and evaluate each of these integrals independently. Starting with the first integral, we have the integral of 1 DX, and the integral of just the constant function with respect to X is just going to be X. So this is the value of the integral for the first part of the sum. Now, for the second part of the integral. We have the integral of 4 X2 divided by XQ + 2. But let's go ahead and move the 4 in front of the integral as a constant. So we have 4 multiplied by X2, divided by X cubed plus 2. One way that we can solve this integral is by using a U substitution. Let's go ahead and allow U to equal to the denominator, which is X cubed plus 2. Then we need to come up with a substitution for DX. We will take the derivative of both sides of our U substitution with respect to X, and that is going to give us DU is equal to 3 X 2 DX. We have an X 2 DX in the integrand, but we don't have this coefficient of 3. If we divide both sides of the DU statement by 3, we will have 1/3 DU is equal to X2 DX. So, by applying these substitutions, we can rewrite the integral to be 4 multiplied by the integral of 1 divided by U multiplied by 1/3 DU. We can move the 1/3 to the front of the integral as a constant, and we will have 4/3s multiplied by the integral of 1 divided by U to you. The integral of one divided by you will simplify to be the natural logarithm of you, so we have 4/3. Multiplied by the natural logarithm of U and by back substituting you into terms of X, we are left with 4/3 multiplied by the natural logarithm of X cubed plus 2, and this is going to be the value of the second integral for the original sum. So, if we take the two values of the sums, and we backs substitute them into the overall sum we were solving, we will simplify the integrals to be X + 4/3 multiplied by the natural logarithm of X cubed plus 2, and because we were working with an indefinite indefinite integral, we will add a general constant C to the end of the sum, and this is going to be the solution to the given indefinite integral. And the overall solution to this problem is going to be B. So, I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.10. ∫ (x³ + 3x² + 1)/(x³ + 1) dx

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2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
10. ∫ (x³ + 3x² + 1)/(x³ + 1) dx