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Evaluate the definite integral fc} + 2x + 3)dx You can use the Fundamental Theorem here_ Your answer should be a simplified expression in terms of a and b. You can ...

Question

Evaluate the definite integral fc} + 2x + 3)dx You can use the Fundamental Theorem here_ Your answer should be a simplified expression in terms of a and b. You can assume that a and b are both positive numbers. Show all of the work clear and complete_

Evaluate the definite integral fc} + 2x + 3)dx You can use the Fundamental Theorem here_ Your answer should be a simplified expression in terms of a and b. You can assume that a and b are both positive numbers. Show all of the work clear and complete_



Answers

Evaluate the following integrals. Assume a and b are real numbers and $n$ is a positive integer. $\int \frac{x}{a x+b} d x$ (Hint: \,$u=a x+b$.)

So we have the general integral X over the square root of X plus B d x Onda. All we know is that A and B are really numbers doesn't really matter what they are. So, first of all, we are going to set you equal to square root of a X plus B Which another way of writing that just as a reminder is a X plus B to the 1/2. So do you deal? Do you over DX If we differentiate that we get do you equal to 1/2 X plus B The negative 1/2 Um, times a remember using the chain rule. Um, And if we solve for X in this case, we have X equals square both sides. So we have you squared minus B divide by a you squared must be over a So, um, let's substitute all this in, um, because in the integral, we already actually have, um X plus B to the negative 1/2. It's just this right here, because it's the negative 1/2. That just means square root in the denominator. Um, so that's convenient. We have our do you there, But remember, if we also want to add a, um, a over to, um to account for this A in this 1/2. Um, in order to subsume d'you for DX, we have to keep balanced by taking a two over a outside of the integral so that when you multiply, it just equals one. So it's even, um, so just substituting. We have two over a times the integral of, um X, which we established is you squared minus B over a times d'you. And we got there not times, do you? But do you, um, man, we got rid of those denominators square root of X plus B because it is a part of do you, So that's convenient. Um, from here, Um, remember, this one over a we have here is just a constant We can pull that out of the integral. So we have to over eight times. One over is it's gonna be to over a square integral of you squared minus B, do you That just becomes too, over a squared times. Um, the integral of you squared is gonna be you to the third over. Three minus the integral of be some constant. It's just gonna be be you, um, plus See, now all we have to dio is substitute all that back into substitute um X plus B back into the use. So we have to over a squared times you to the third. So that's, um root X plus B to the third, um, or we can write that as a X plus B to the 3/2. Because you know Bruto ax plus B is just, um X plus B to the 1/2 power over free minus b times a X plus B times X must be to the 1/2 because you equals excellent speech of 1/2 plus c. And so, no matter what A or B is Noah, as long as they're real numbers, this is the general formula for solving that integral.

Problem. 74 Integration off export and sine inverse X The X, which is equal to U, is equal to sign Angers X the you is ableto B X over one minus X squared. Needy is equal to export and the X the is able to export and plus one over and this morning. So this is equal to sign Angers X and yes, times export and plus one over and plus one minus generation of X. And that's one over and plus one times he x over square root off Omanis X squared, which is equal to export any plus one times. Signed English X over and plus one minus one over M plus one figuration off X n plus one over square root off one minus X squared years.

The given integral is integral off A X square plus B X plus C D X. So we will start by first breaking this integral into three parts integral of a X square. The X plus integral O B X D X plus in two G L O C D s. Since A B and C are constants, we can take them outside the industrial side. A integral of X squared dx plus be integral of X bx plus c integral of dx. Upon solving the needles, we get a multiplied with X cube divided by three plus. Be multiplied with extra square divided by two plus C X plus capital C. Which is a constant of integration. We can also represent it as C. One so that we can differentiate between the small C and the capital C. Upon further simplifying, we can rewrite the entire expression as E X cubed divided by three plus the X square divided by two plus C. X plus capital C. One. This will become the final answer.

So we have this abstract integral, um, axe times A to aid a expose B to the end Power DX. And all we know is that A and B are any rial number and n is some positive integer. So if we use a u substitution where we have u equals a X plus B such that d u is equal to just a times DX, we can substitute for, um, use instead of exes. So remember, if we want to multiply the inside of the integral by A in order to get the a d. X in order to replace on that with do you we have to multiply the outside of the integral by its reciprocal one over A just to keep it even. So, we have one over. A integral X is just If we use this equation to solve for X, we just have, um you might as be over a so u minus B over a times you to the end power. Do you, um, now, here again, we can take out this one over a because that's what's gonna be a constant. So we are left with one over a squared, integral you to the end times u minus B do you? And if we distribute this you to the end two u minus, B um, we get one over a squared integral of you to the end plus one because that's just exponents rules you to the end times you is just you add up the exponents u to the n plus one minus b you to the end, Did you? And here we can just solve these integral separately. So that is equal to one over a squared times You the integral of you to the, um and plus one, um, at another ah one to the expo Nene of n plus two over, um, and plus two, um, yeah, over and plus two minus be times you to the M plus one over and plus one plus scene. And if you, um, substitute the use back in for X plus B, that just comes out to one over a squared times a X plus B to the end, plus two over and plus two minus b you to the end, plus one over. Um, sorry. B Times X plus being. That's what NU equals to the M plus one over m plus one plus c. So no matter what a b or n are as long as they're real and and is an integer a positive instrument, um, we can use this general formula that we just saw for to find the integral.


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