5

5, Integrate: X +3x4 dx 2+...

Question

5, Integrate: X +3x4 dx 2+

5, Integrate: X +3x4 dx 2+



Answers

Perform the indicated integrations. $$\int(x-2)^{5} d x$$

Even function is integral of two x rays to the power Five days. Now we have to find the indefinite integral here. We can see that it is of the form integral of x rays to the power N. D. X. Which can be written in one, divide by N plus one. Multiplied by exodus to the power N plus one plus. See now apply this rule. So our function integral of two x rays to the power five V. X. Can be written as one, divide by five plus one. My deployed by x rays to the power five plus one plus C. After simplify the get here. And this is to to divide by six, multiplied by x rays to the power six plus six. Now can sell out we get x rays to the power six, divide by three plus C. And this is our final answer.

Yeah. In this solution that you want integrally studies. Okay integral one divided by X squared plus two X plus five D. X. Now in the community is saying that solved this given integral. So let's start answering this fusion. So the given integral is in the hospital for style. Right. The given integral that is integral. One divided by X squared plus two weeks plus five D. Eggs. No, it can be simplified by doing some algebraic operation. Algebraic manipulation in the denominator of Inte Grant. So it will be written as that is integral. One divided by X square plus two X plus five will be denied status one plus four B. X. No simplified it. The between integral with the sitting on right side. So it will return as there is Integral one divided by after collecting these three terms together. Then it will be denies. That is X plus one. Holy Square plus. Now remaining histories. Let's or I can say that it will be regionalized. There is plus two is square li X. So you can see that the Now the given integral is written in a simplified forum. So it means my next tv's. I have to find the answer for this simplified problem integral which is written in a simplified forum. So here I will use a method that is called substitution whether yeah. So after finding this method, sorry, after finding getting the integral which is written on the right side or I considered simplified for him, then I will get the automatically answered forgiven integral. So here I am using a method that is called substitution matter or assuming then right as you assume that X-us one is equal to you not take a differentiation on both sides. So it will be done as there is DX is equal to do you. So I can say that this is a step number one and this is a step number two. Yeah. Okay. Yeah, This is a stream number two And uh this is still number three. So by using this Eastern number two and number three the integral which is in the simplified form. In Eastern number one will be written as that is integral. D X divided by X plus one. All the square plus two is square is equal to integral. Dx will return is there is by using this easter number three? I can say that. Do you divided by X plus one? Will be recognized by using this easter number two. I can see that you square plus two square. No next step is yeah, I'm using a Eastern integration formula. That is formula is integral. Bu divided by a square plus you is square is equal to one over a 10 universe. You over a plus C. Turn inwards of you're a plus C. So use this integration formula and put the value of A. That is equal to to find the integral which is sitting on right side. That is integral. So I will say that after using this formula and uh substituting the value of it that is equal to two. Then it was written as result will be that is left side will be written as it is the same. It means the integral in simplified form will be regional studies integral D. X divided by X plus one. All the square plus. Do you square? Yeah. Is equal to indication of one divided by U squared plus two square about. You will witness that is One divided by 2. 10 universe. You divided by two. Let's see. Yeah. No. Next step is for the value of you from step number two that is used equal to X plus one. So I will get the answer in a get the desert in original acceptable. So I considered integral. D. H. Is the, sorry D. X divided by X plus one. Only square plus two squared is equal to one divided by two dozen universe X plus one divided by two and myself X plus one divided by two plus C. No, I can say that the answer forgiven integral. That is after writing the given integral on left side because this is a simplified form of Cuban integral. So in the answer I was writing the human integral that is when divided by X square plus two X plus five Ds. You can so you can also use this strip number one Ladies. You can see that the given in this integral, which I get in a box tape is simply by forum of this given integral. So I can see that the ourselves forgiven integral that is integral. One divided by X squared plus two X plus five G X will be equal to one over to 10 universe of X plus 1/2 plus C. So this is the answer for given fusion. Thank you.

When you're asked to find the integral of one over two X. Plus five, you really do have to be careful. Um Because I am an advocate for students doing math in their head because you see that it's one over a function. So then the integral must be natural log of that function. Uh You need to have a plus C. In here. But the real issue that we would have is that the derivative of this would include the chain rule. So you'd have to multiply, let's write this in blue. Multiply by the derivative Of to expose five. Which would be too well that's not over in this answer. So what you would really have to do is divide by two or multiply by the reciprocal. Um So I have a lot of students that do that math in their head. If you're not quite ready for it, you could use U. Substitution which you've learned in a previous section let U equal the denominator. So d. You would equal to two D. X. And what I usually do with my students is have them divide that to to the left side. So when you go to do this problem, I'll switch back to black uh would be the integral of one over you And then DX is equal to 1/2, do you? And you can move that one half in front. So then this and enroll in equal one half natural log of. You don't forget about your plus C. And as I mentioned before, we let U. Equal to X. Plus five. So this is a step by step explanation as to why there's a one half in front. And you can double check that your correct by taking the derivative of our answer and the derivative of our answer needs to equal the starting problems. And it does. Yeah.

Let's evaluate the following incident, so we should use partial fraction decomposition here. That's the section that we're in. So looking at that denominator, we'd like to know if that factors we have a one Ex clears two X plus five. So we see here are A B and C one, two and five. So let's look at the discriminated B squared minus four a. C. So be square is four minus four times one times five. That's a negative quantity. So that tells us that this denominator will not factor over the roll numbers and therefore this fraction that were given in the Insel girl. This already is a partial fraction, so we're ready to integrate. And one way to integrate this is they're just split this into two parts, two fractions. And before I do that, let me actually go to the side and complete this where X squared plus two x plus five. I take half of the two in front of the X. It's clear that and that in there, and then I do five and then minus one to make up for the adding one. So we had X plus one square plus, for which we can write is too square. So that will be our denominator x over not necessary to break this into two in a girls. But it may be easier for you. Otherwise you can just leave it in there and try it that way, Not that much different. We'LL pull out that for and both of these in a girls are very similar same denominator and they both could be solved using the troops of So for now, for convenience, let me just refer This is integral A and will be So let's look at a furs for a we could see that actually will both have the same tricks up here. We should take X plus one to be too thin Data then DX to see cans where and so we can write a So coming up in the numerator with C Just an X there so you can solve this for X. So subtract one from the side We get to ten minus one and then we have the DX. That's two sea cans where and then on the bottom we see the X plus one is two ten. So we swear that so four dance where data plus four. Let me factor out before there. And we know that tan squared plus one and see Can't square so we could cancel those. And then we have We're left over with two ten data minus one in the prentiss is and then we have to over for this is the half So we can write. This is just the integral Santana minus the half and then in a girl's hand, naturalized absolute values he can if it might help you here to integrate. You could also do a use up here, let you be scientific and then write Tanja and signed over co sign or little Excuse me here. You should do it. Youbecause I Taylor, come on into a u substitution. Otherwise you might memorize it. And integral of negative on half. That's just negative. State over to Don't worry about this constant C i'LL add that later on at the very end. So then here, let's rewrite this is natural log And then, actually I'm running out of room here, so I have to go to the next page. But the last thing to do here is to rewrite both of these in terms of the original variable X. So let's go to the next patient. I'm running out of room here, So using our trick sub, we have a right triangle. So recall our substitution. So this means that X plus one over to his tan and that's will give us the triangle. So Tangent is opposite, Divided by Jason and then using the category dirham. We have the high partners, but she could write as X squared plus two x plus five original denominator. But inside the square. Okay, so going back to the previous page, we had natural log absolute values he can't and then minus Saito over too. And now we can evaluate this. So that's natural log with absolute value and then to evaluate the sea can it'LL be high pardons over Jason So that's X squared two X plus five it in the radical. Then divide that by two and then minus and it's a soft earth Ada. We just take our tan on both sides of this equation. Here, data equals tan in verse X plus one over two. So that's a plus one divided by two and so and also we have a two here on the bottom watch out for that too. So we have tannin, verse X plus one over to and also divide that by two. So this is our answer for the first Integral A. The second integral role required that much work because it's basically they're going to use the same tricks over that we already used. So using the same substance, the same tricks up that we already have for part B. So that was four times the integral X plus one squared plus two square the ex weaken right, this is for and then the ex we recall that's from the previous page and also on the denominator. We saw what happened on the previous integral that will become for she can't afford it. So now it's We just go ahead and simplify this so we could cancel and we get out of two and the sea can square terms will cancel data. She's to theatre. So too, ten members X plus one over two. So that's our answer for part B. Now the final step decisions at these answers together. Let me go to the next case to do that. So add a plus me So we have natural log no need to write absolute value here anymore because the thing that writing on the inside if these are positive numbers positive on the top and then positive on the bottom. And we had a two down there. So actually, let me rewrite that Using the law properties Ellen a over B someone I'll do my best to make This answer matched the answer in the book. So using that property, I'll pull out the Ellen too. And then we also had minus arc tan X plus one over two, divided by two So that was from a and that for me And then our constancy of integration the last up here sitges combined light Serbs we can combine these arcs dancers So we'll have three over to our ten and then we'll also have this lottery is, um if we want we could also use another property logarithms Let me write that over here Ellen Ate of the bee is be hell in there we can use this property We see that there's a radical That's a one half power So let's pull out that one half in front of the dog and they're technically we should use absolute value that may be negative. And then this is a constant Seize the constant so luscious. Add those constants together and quality. So here, by deed, I just mean C minus Ellen, too. That's just a constant, and theirs are finalized.


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