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(10) Solve the integral equation9)8 f(t)=l+t+ 2p.(2- 2)(2) f [ 3...

Question

(10) Solve the integral equation9)8 f(t)=l+t+ 2p.(2- 2)(2) f [ 3

(10) Solve the integral equation 9) 8 f(t)=l+t+ 2p.(2- 2)(2) f [ 3



Answers

$7-38$ Evaluate the integral.
$$\int \frac{10}{(x-1)\left(x^{2}+9\right)} d x$$

Days did it off, then a bone X minus one whole Esquire into its squad plus nine WinDix. No, this can't be done. As done Indeed. The love. It's minus four off home. 50 Indu. It's far less nine, my ass. What about 52 x minus? Bless. What about down X minus one? Disquiet dicks. This equal Do whenever I fight. Do you love X minus for a bone? This is while plus nine d s, minus one upon fired. Do you love dicks? Upon X minus one plus in deal off, dears. Upon X minus one. Willis. What? So this becomes one of one fight. Do you love x upon? It's a smart plus mine. DX minus full indigo DX upon it is about blast. Mine minus went off on five. Ellen. More x minus. Who wanna glass X minus. Wanted about of minus one upon minus one plus C. Now you ever saw X the X upon its squared plus nine. So this becomes I want a bonfire and do, huh? Ellen Moore. It's a squad. Yes. Nine, my ness. Fool Upon three Dying waas X by three minus one of find and then more X minus one minus one of our own, and this is X minus one plus C. Now we love the dealer guns, that is. What about Ellen Mortal? It's a squat plus nine minus four upon 15 time Waas ex upon tree minus one aboard flight. Ln more X minus one minus one upon X minus one. Let's see. This is the final result, and it is the answer still dropping.

In this fusion that given integral is that is integral. The X divided by nine X squared plus 18 X plus 10. So this integral is given. And now in the question it is saying that it's all over this given integral. So let's start answering this fusion. So in the answer first I will write the given integral that is integral, B X divided by nine X squared plus 18 X plus 10. So it will be written as it is in a year. I will try to write in simplified forum of Cuban integral. So I can say that it will be written in studies integral. The X divided by nine X squared plus 18 eggs plus 10 will be returning status plus nine plus one. Mhm. No again simplify it. And I can see that integral. D. X divided by nine X square plus 18 X plus 10 is equal to integral D. X. Divided by after collecting these three tons together. Then it will return to studies three x plus three. Holy Square plus no plus one will be written in stories plus one square. No, I can say that. Look this step. So I consider this Cuban integral is now change in simplified forum which is sitting on right side. So no next step is I will try to find the answer for in google which is written on right side. So this is a step number one. So in the next step I will use a method that is called substitution method. So and assuming that act three X plus three is equal to you. So now take a differentiation on both sides. So it will be general studies. The U. Is equal to sorry three DX. is equal to do you. So I can say that this is a step number two and this is a step number three. So by using this step number two and number three. The step number one will be written in studies. Mhm. Yes integral D. X. Divided by nine X square plus 18 X plus 10 is equal to integral. DX will be written as status by using this eastern No. I can say that Do you over three and one divided by three X plus three. All square will be written is by using this issue number two. I can say that there is three X plus three is equal to you. So it means one divided by you square plus one square. So I can say that the given integral is now changed into variable by using this safety tuition matter. So that is in the result I can see that B. S divided by nine square X plus 18 eggs Plus 10 is equal to 1/3. Well come outside. So remain steady state is integral. Do you divided by U squared plus one square? So my next step is I had to find the interrogation of U squared plus one the square about you. So for this I will use the standard integration formula. That is integral. Do you divided by a square plus you square is equal to one or a 10 in words of you over a plus c. So use this standard formula in a a boy step and put the value of A. That is equal to one. Then I will write the result. I will get the result that is equal to look the bush tape. So I consider that the upside will region. Is that the same? That is human integral. That is integral D. X. Divided by nine X squared plus 18 X plus three. Sorry plus 10 is equal to no right side. Will written as studies 1/3. And the integration of you square plus one square about. You will be written by using this external integration formula and putting the value of A. That is equal to one. I can say that yeah In the bracket the term will be eaten as there is one Over one that will be one and one multiplied by 10 universe. You over a everybody that is one. So I can say that 10 universe you plus c. No next to bees remove the bracket and for the value of you that is equal to three explosive. It is written in escape number to that is use equal to three X plus three. So so shoot this you is called a three X plus three in a boy step. So dessert will be eternal. Studies integral. The X divided by nine x square plus 18 X plus 10 is equal to 1/3, 10 inwards of three x plus three because See over three. So the answer forgiven integral is that is integral, B X divided by nine X squared plus 18 X plus 10 is equal to 1/3. Then in worse off three X plus three. And then plus I considered that C +03 is any question value and I'm assuming this By any question time that is c. one. So this is the answer for a given integral. Thank you.

To integrate the integral of D. X. Over the square it of nine X. Great plus six X plus 10. Not that we can rewrite this into the interval of one over square it of nine times X squared plus two thirds X plus 10 dx. And then from here we have the integral of one over the square it of nine times X squared plus 2/3 X plus 1/9 by completing the square. And then plus then minus one D. X. Which is the same as the integral of one over the square root of nine times the square of expose one third plus nine dx. And then from here we apply substitution. We let U equal to expose one third so that D. Is just the X. And from here we have the integral of one over the square root of nine U squared plus nine D. U. And can factor out the night inside the radical. And we have the integral of one over the square root of nine times the square root of U. Squared plus one D. You okay next we apply Trigana metric substitution. We let U equal to tangent data and the U. S. Just speak in squared theta. D. Theta So we have 1/3. And a girl of one over square of tangent squared theater plus one time second squared era Diferente which simplifies to one third. In a girl of second squared theta over the squared of second squared data. D. Theta or this is just 1/3 integral of sick and sarah D. Theta. Now this is equal to one third. Ellen absolute value of seeking thera plus tangent data plus C. But since there is not the original variable we go back to our substitution that if you is tangent Ceta then if we draw a triangle and paris in here we have tangent terrace opposite over the adjacent side. So to be you and then one here and this will be the script of use great plus one. So seek and terra would be hypothesis over the adjacent side. That's just squared of U squared plus one. And so in here we have one third. Ellen absolute value of the square. The U squared plus one plus you. And then plus C. Now since you is excellence one third. This is the same as one third Ln uh the square it of X plus one third squared plus one plus X plus 1/3. And then plus C. This is always positive. So the absolute value disappears. And then from here we can simplify this into 1/3 L. And of the square of X squared plus two thirds X plus 10/9 plus we have experts. 1/3 plus C which is equal to one third Ln. Of the square worth of nine X squared plus six X plus 10 plus three X. Was one and then plus C. And so this is our indefinite and a grill.


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