5

1) (10 points) Evaluate: 4x +18x2 _ 12x2 +36x +5 dx...

Question

1) (10 points) Evaluate: 4x +18x2 _ 12x2 +36x +5 dx

1) (10 points) Evaluate: 4x +18x2 _ 12x2 +36x +5 dx



Answers

$7-38$ Evaluate the integral.
$$\int \frac{x+4}{x^{2}+2 x+5} d x$$

Again this question. We have to solve integration. D x divided by X access to the power four plus one. Okay, so we will rewrite our question as integration D d x divided by x. Okay. And we will take access to the power four comin out of this. It will be access to the power 41 plus access to the power minus four. Okay. Or we can say it will be d X divided by X raised to the power 51 plus X raised to the power minus four. Okay. And now we will take this portion has substitution. You okay? You will be one plus X raised to the power minus four. Or we can say do you will be minus four X raised to the power minus five dx OK, or it will be. Do you divided by four and minus will be one upon express to the power five dx. Okay. And this is presenting our portion. So it will be this d x, divided by express to the power five will be minus d'you divided by four. Okay. And this will be you, so it will be even upon you, okay? Or we can say our question will be minus one upon four. Integration do you divided by you and its integration will be minus one upon four. An integration of and by you do you will be learn You plus see Okay. And now we will back. Substitute the value of you That is one plus X raised to the power minus four and it will be minus one by four. Okay. And Ellen, one plus access to the power minus four plus C or we can say it will be minus one by four. Ellen, it will be one plus one upon express to the power four plus c. Okay. And to get it simple, it will be minus one by four. Okay, island, it will be expressed to the power four plus one, divided by express to the power four plus c. Okay. And we will have We have to eliminate this negative sign. So this will be won by four Ln It will go to the universe of the power of X. Okay. And it will be upon We will be be upon a okay, and it will be access to the power four divided by access to the power four plus one plus. See? Okay. And there is no need for the absolute value here because express to the powerful is a positive value. And this will be our final answer. Thank you.

That you can untangle can be Internet. Fourth in between, it's two on toe. The paradox Last 10 came between two and five with our techs went in with the Power X the X, which is for 20 years ago to pick up our X last five of the power X The X things gives us goto paradox and then to plus I put about by Ellen to bless you should never around.

Again this question here to solve integration. X Plus two divided by X square plus three x minus four DX and the limited to and upper limit is four we can rewrite. Our motion is limited to four x plus two. Do what I did by X X plus four and X minus one DX OK, or we can say we can write to before it will be a divided by X plus four plus B, divided by X minus one and DX. OK, now we have to find the value of A and B here, so we will compare this. X plus two divided by X plus four X minus one equals to a divided by X plus four plus B divided by X minus one. Okay. And by comparing, we will get X Plus two equals to a X minus one plus B X plus four. Okay, so by the comparing coefficient of X and the constant, we will get a plus. B equals to one and minus a plus four B equals two to Okay, this will give us the value of a N B. A. Will be two by five and we will be three by five. Okay, so we can rewrite our question. That is, this we will rewrite is limit limit 2 to 4. Okay. And a divided by X plus four that is too divided by five. Okay. And x plus four and plus three divided by five. And it is X minus one. Okay. And be X. So the integration of this will be two divided by five and integration of an upon X plus four will be l N X plus four. Blessed three by five Ellen X minus one and limited to upper limit is four. Okay. And now we will apply the limits here and first upper limit. It will be two by five Ln X plus four. That is four plus four. It will be eight plus three by five. Ln four minus one. That is three. And now? Yes. And the lower limit that is too. It will be to buy five Ln two plus four. That is six and plus three by 5. 10 to minus one. That is Ellen one. Okay. And now it will be Dubai five. Okay, two by five. And this too can be in the power of eight. And it will be one by five Ln eight raised to the power to Okay, this can be written like this and this can be done. Has won by five Ln to be raised to the power theory. Okay. And this can be written is minus the salary to buy five and l n six. Same. Sorry. This will be won by five. Okay, one by five. And l n six raised to the power to okay. And Ellen, one is zero. So now we will take one by five years common. And it will be Ln Okay. Here the square plus Ln today's cube. Okay. Or we can say sorry. Ellen. A ceasefire plus Ellen, treat you okay. And minus Ln 60 square. Okay. And now we know the property. L n a plus l n b will be Ellen A or B and Ellen K minus. Ln we will be Ellen. A divided B. Then it will be won by five. And Ellen, it is square 0.3 square three Q and divided by 60 square. Okay. And it will be won by five. Ln and when we saw it, we will get 48. And Ellen, 48 divided by five will be our final answer. Thank you.

We want to integrate this function recognizes the improper function. So let's perform long division. Exhale. Yeah, Put a minus sign. We get minus four eggs less full. So really, we can rewrite this as thanks. Minus four X over X squared plus four plus full. Over. Explain this for I want to use some forms for this one. I want to try and fit into this form so I can't use the one motor ethics for this one. I want to try to feed into this bomb. Yeah, it was one of the a dungeon. Impressed. That's over. A thistle is the Yeah, so you can see that if I let FX be, I squared. Plus for my entrance B two x so I can write this fall. Yes, I can split it into so explain putting Children. Yeah, and for this peaceful Okay, we write it as two square. So I can 50 and do this one here where my is three. So now that I have this form, I can write my answer ass integrating ex. I'll get it's square over to integrating this portion. I'll have minus two fitting the form will be long mark off X squared plus sport now, but I want to highlight the X squared plus four will always be positive, so we do not need to write more. We can write it as a second, and for this term it will be plus for 15. With this one, it will be one over to tension in verse X over to So our final answer is X squared over two man is too long X squared plus foreign break it plus two tension in verse X over to Don't forget to put your plus c


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