The integrals in Exercises 1–44 are in no particular order. Ev...

Question

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.

∫₋₁³ (4x² - 7) / (2x + 3) dx

Accepted Answer

Hello. In this video, we are going to be evaluating the following definite integral. Now, we are given the integral from 1 to 4 of 2 X2 + 3 X minus 5 divided by X minus X + 2. Now, if we take a look at the integrand, the power of the numerator is different from the power of the denominator by exactly one value. And in fact, this integrant is going to be top-heavy because the numerator has a higher power than the denominator. So what this means is that we can simplify this integrand by using polynomial long division. So let's go ahead and set up the polynomial long division to simplify the integrand. We will have x + 2 divided by 2 x 2 + 3 x minus 5. Now we want to think of a number. That we want to multiply to the first term of our denominator to try to match the first term of the quotient. Now here, we can multiply X by 2 X, and multiplying X by 2 X will give us 2 x 2. But if we are multiplying the first term of our denominator by 2 X, we have to multiply every term in the denominator by 2 X. So 2X multiplied by 2 will give us 4 X. Then we have to subtract. This this product afterward, but in order to subtract, we have to distribute this negative sign to each term. So that is going to change the signs in each of our products. So we will have -2 x 2 minus 4 x. Now if we combine like terms, the two x 2d terms will cancel each other out, and what we are left with is 3 x minus 4 x, which is g x, and we will bring down the next term, which is -5. Now we can match X to X by multiplying by -1. This will give us g x minus 2. But once we subtract and distribute the negative sign, this will change to positive x + 2. This will cancel out our variables x and leave us with -5 + 2, which gives us a remainder of -3. So how does this simplifier integrate? Well, now if we rewrite the quotient as the integrant, we can rewrite the integral to be the integral from 1 to 4 of 2 X minus 1 minus 3 divided by X + 2. Let's go ahead and integrate term by term. Now 2 x is going to integrate to just be x squared, and -1 will integrate to be x. Now these two terms are going to evaluate from 1 to 4, but integrating -3 divided by x plus 2 is a little trickier. If we take a look at that integral independently, we have the integral from 1 to 4 of -3 divided by X + 2. Now what we want to do is simplify this integrand by using a U substitution. We will allow you to equal to x + 2, then DU will equal to DX. So, once we apply the substitution, we can rewrite this integral to be the negative integral of 3 divided by UDU. Now, because we're applying a U substitution over a definite integral, we want to plug in our boundaries for X in our U substitution. Doing so will give us the new boundaries from 3 to 6. So, once we evaluate this integral, this is going to give us the integral of -3 multiplied by the natural logarithm of U, and this will be evaluated from 3 to 6. Evaluating this integral is going to leave us with -3 multiplied by the natural logarithm of 6 minus -3 multiplied by the natural logarithm of 3. Now if we were to simplify here. We distribute our negative sign, which will give us a sum. And if we factor out a -3, we will have -3 multiplied by the natural logarithm of 6 minus the natural logarithm of 3. And by using our natural logarithm properties we can combine the natural logarithms as a quotient and we will be left with -3 multiplied by the natural logarithm of 2. This is going to be the value of the last integral. And now all we need to do is we just need to go back and evaluate X2 minus X from 1 to 4. That will leave us with 4 squared which is 16 minus 4. -12, which is just 1 minus 1. This second term will simplify to be 0 and 16 minus 4 is going to simplify to give us 12. So this is the value of the first two integrals evaluated from 1 to 4. And now, if we combine the two results together, our final solution is going to be 12 minus 3 multiplied by the natural logarithm of 2. This is going to be the solution to the given definite integral. And the overall solution to the problem is going to be option A. So I hope this video helps you in understanding on how to approach this problem, and we will go ahead and see you all in the next video.

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.∫₋₁³ (4x² - 7) / (2x + 3) dx

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The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫₋₁³ (4x² - 7) / (2x + 3) dx