5

Point) Note: You can get full credit for this problem by just answering the last question correctly: The initial questions are meant as hints towards the final answ...

Question

Point) Note: You can get full credit for this problem by just answering the last question correctly: The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit:Consider the indefinite integral x (3 + Ix" ) dx Then the most appropriate substitution to simplify this integral is u =Then dx f(x) du where f(x)After making the substitution we obtain the integral g(u) du whereg(u)This last integral is: +C (Leave out constant of integr

point) Note: You can get full credit for this problem by just answering the last question correctly: The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit: Consider the indefinite integral x (3 + Ix" ) dx Then the most appropriate substitution to simplify this integral is u = Then dx f(x) du where f(x) After making the substitution we obtain the integral g(u) du where g(u) This last integral is: +C (Leave out constant of integration from your answer:) After substituting back for U we obtain the following final form of the answer: +C (Leave out constant of integration from your answer:)



Answers

The indefinite integral can be found in more than one way. First use the substitution method to find the indefinite integral. Then find it without using substitution. Check that your answers are equivalent. $$ \int 3 x^{2}\left(x^{3}+1\right) d x $$

Again this question. We have to solve integration to X into the bracket Access where? Minus one DX. OK, so as part of the question, we have to solve it first of all by using substitution method and then without substitution. Okay, so first, with substitution, you'll substitute U equals two X squared minus one, and it will give us the week was 22 x dx. Okay. And now integration. This excess square management will be you. And these two x dx will be our d You okay, So the integration of you do you will be you square by two plus C or we can say u is X squared minus one. Then it will be X squared minus one Holy Squire by two plus c So this is the answer by using substitution method and now, without substitution, we will. It is two x and into the bracket X squared minus one DX. So we just multiply these two x and two x squared minus one. So it will be two x cube. Okay, two X cubed minus two x and D x. Okay. And by the integration, it will be two x rays to the power four divided by four minus two x squared by two plus c. Okay. Or we can say it will be X raised to the power four divided by two minus X squared plus c. Okay, so this is the answer by without substitution. Okay. And now we will check. Both of these answers are same or not, or equal or not. Okay, so left hand side, the first answer that is X squared minus one will square divided by two plus C and the right hand side X raised to the power four divided by two minus X squared, plus c. Okay, Now we will check if these answers are same or not. So we have taken question mark in between them. So first of all, we will multiply both of these side by two. And it will be X squared minus one Holy square. Okay. And plus, it will be to see that can be written again. See? Okay. And it will be expressed to the power four and minus two x Esquire. Okay. Plus, to see that can be again. See? And now we will expand the left hand side. Okay, We will. In the left hand side we will expand the square and it will be expressed to the power four. Okay. Plus one minus to a B. And it will be minus two X square plus C and question mark and access to the power four minus two X squared plus C. Okay. Here one is constant at sea is also constant. So we will miss them. Okay. And it will be expressed to the about four minus two x squared plus C and right inside access to the power four minus two x squared plus c And we see that the both sides are equal. So we make them equal and we can can grow you can You can do the integration by any method. And the answer will always be same. Thank you.

In the human problem. We have ghost statement. Beauties deal off singing last takes, beings. So this is very generous. Save your thanks. Indigent, the ex. My medication, Dear song. Second worst sticks. What is this? This is nice. Second or six into sex, my ass. In addition off. You know what? It's a minute you do this. Delis. This gas is each other. This becomes takes second or six, minus divisional and just No, this guy's chain we have. Great. Except for us. Saved you, dear sequence. Say you, uh, you days second rustics. It's bus you. So this is even. It's saving most. Fix my conviction off. See you. Damn. Do you know where expire? You minus So this is giving us Thanks. Saving versus sticks. My distribution off. See you. Do you want 10? This Kansas Each other becomes Thanks. Staking your six minus in division off. Say you do. No, this is even ask Stake in worse. Fix my hands and then can you guess? Thank you. Let's see. This is evil Ex saving your six minus Ellen Mar. Uh, second Worse. Yes. No. Save the sinking. Worse. See, you say whose ex one plus x Let's see. This is given as it's second. Was Nick's more. No, thanks. Let's see. This is that family. So the

We want to find the most, generate a derivative two times eat next minus three times. It's made of two works. So let's first go to imply the linearity of the entire derivative so we can distribute this across the subtraction. So we get these two and trolls here, and then the next thing we can apply for the linearity is that we can pull out any Constance will be two times integral of the ex, minus three times the integral of E to the negative Two ex the ex. Now, each of these will follow directly from our anti derivative started to give us the 1st 1 K use just one. So they're just gonna be each of the two x than minus this next one. This is where Kate is negative too. So it should be won over negative to heat of negative to X and then plus constancy. So we should have two different constants, really, that we're adding together. But if we had to Constance, we just get another constant. Those go ahead in some quiet this town street too. He to the ex that these negatives here will cancel so they'll be plus three house. He too. Negative two x I'll see in this here will bear most general anti derivative

I want to find the most general anti derivative for X plus one. So this will fall into the case if we look at the anti derivative chart that they give us being exit in for each of these. So let's first use the fact that the anti derivative or the integral is a linear operator, which means we can distribute this across plus signs so we could rewrite this as the integral of X plus the integral of one and now integrating each of these while X is really just X to the one so we can write this as X to the one plus one, and then we divide by that new power which is going to be, too. And then we could think of one is really just X to zero. So this is another case of kind of power rule for integration, but we could also just think about it as well. It's a constant, so that should integrate to just X, and we need to add our constant. See to this. Let's just go in and clean this up close of this should be X squared over two plus X plus C, and this hair would be the most general anti derivative


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