evaluate-the-integral-int-frac1xaxb-d-x

Question

Evaluate the integral.
$$\int \frac{1}{(x+a)(x+b)} d x$$

Fuzail Shakir · Accepted Answer

Okay, so if we decompose it into partial fractions, we end up with a over X plus small a plus V over X plus small. Be multiplying both sides by the least common denominator. We end up with a multiplied by X plus V plus be multiplied by X plus a so setting X is equal to native A. We end up with a is equal to one over B minus eight on setting X is equal to negative. B we end up with B is equal to one over a minus b So putting this back in we end up with won over B minus a times the integral off one over X plus a minus one over X plus b. So using the law, properties get up with one over B minus a times ln of the absolute value off X plus a divided by explosive speed plus C.

J Hardin · Answer

let&#x27;s evaluate the integral of one over X plus any times X plus B because Andy or could be any real numbers, this problem will depend on whether or not and your equal, because if a equals B, for example, this means that we can write the general as one over X plus a square. And that will change how we go about the partial fraction to composition. This will be something that the book will call case too. But really, for us in this case, this basically is already in the form of a partial fraction to composition. We just could actually go ahead and integrate this with the use of used the power rule. So this is why the case, it matters whether or not a equals B and we&#x27;LL see in case, too, when a does not equal b we&#x27;LL have two distinct linear factors in the denominator. So that&#x27;s a different looking partial fraction the composition what we have over here now we&#x27;re just circling right now. So first for in this case, a equals B we rewrite integral and then you may be able to integrate it how it is and once that. But if this plus a is bothering you just go ahead and do it yourself. Then you get do you equals the X. Okay, so we could write this integral is one over you square, do you? And then you could use the power rule to get negative one over you. Plus he and then go ahead, replace you with X plus a. So, in case one, if a equals B, this should be the general. Now we have to also look at the case in which there aren&#x27;t equals. Let&#x27;s do this next. So that&#x27;s case one all over here. Boxed off in green. Now for case, too. And you&#x27;re different. Distinct, not people. So now we can go ahead and take the original Fraction one over. That&#x27;s plus a this this form right here as its in with A and B distinct. This is not a partial freshen. The composition here will be in with the book calls Situation one or case one it went. The denominator has this thing&#x27;s in factors that are linear. So we&#x27;ll have capital a over explosive plus capital B over X plus little bee. Now, looking at this latest equation, let&#x27;s circle is in black. Let&#x27;s go ahead and multiply both sides by the denominator on the left X plus little eh X plus little bee. Good and multiply both sides by that. And we should get one on the left capital. A Times X plus little bee plus capital B Times X plus little and the right. Let&#x27;s fact, throughout the X on the right hand side, we have a plus B, and then the constant term is capital a time. Small B plus capital B time. Small eh? Now, on the left hand side, we see that there&#x27;s no ex term. This means that the ex term, the coefficient of the extreme on the right must be zero. And that&#x27;s equivalent to saying B equals minus a. And then, since the constant her among the left is one. That means that the constants room on the right also has to be one. And now, using the fact that B equals minus A, we could come back to this equation up here and saw for large, eh? So it&#x27;s fact throughout a large a Here and divide we get one over little B minus little, eh? And then, since being equals minus a due to this equation Over here we have negative one over B minus some running on room Here, let me go on to the next page. So the next thing that will do it before we go on is we&#x27;LL take that. We&#x27;re ready to integrate We&#x27;Ll take this original problem here. We wrote it in partial fraction of composition and now we just found a and B So it&#x27;s about unplugging these values for Andy into the perfect reaction to composition. Yeah. So this means that the original problem Let&#x27;s write that out. And then we found large, eh? Which is one over B minus a. That&#x27;s the constant capitally over explosive A And then we also found these Hey, B was negative one over B minus A. Oliver explodes be. And this these terms are easier to generate on the first integral and might help you to do it. Use up U equals explosive. For the second one, you can go ahead and take new equals X plus Meet Teo. Integrate these And when we integrate, we should have won over B minus a natural log. Absolute value of you are explicit and then we have minus one over B minus A for the next integral. That&#x27;s minus is coming from this term up here and then natural log. Absolute Value X plus B. That&#x27;s our integration. And then, since this is a definite over all week putting to sea, the last thing to do here is just to simplify. We see that we can pull out a one over B minus a. From the first two terms. You have Ellen X Plus a minus Ellen X plus me, those air an absolute values and then the last thing we can do. Go ahead and use your log properties to rewrite this determined Prentice&#x27;s. We have won over B minus a natural log, absolute value explosive over X plus me plus see, and there&#x27;s a final answer.

Audrey Fong · Answer

we want to integrate one plus x over one plus X squared. Let&#x27;s read it for us are 1/1 plus X quay plus X over one plus x quay. The X now we know for 1/1 plus x square with integrated work engine Inglis X So that&#x27;s fine. So for these, we were use at Time X over ethics trying to fit this bomb. If you can fitness home our use the result of non model X Let&#x27;s see. So that&#x27;s trying that have exploded denominator, which is one less x squared when we do friendship Ruegen two x So I&#x27;m looking over to over here. So this is what I would do. So for this one, he would just beat engine. It was X time for the 2nd 1 I would put they have here so I can put too X over here. Now my numerator is have prime X might, you know, many days, FX. So I can use this result here So I can really put along 11 plus x ray. Let&#x27;s see what sees a constant now since one plus exquisite, always positive, we can remove them on so final answer is tension in verse. Thanks less, huh Long? Just a normal break It one plus x squared. Close. Break it. Let&#x27;s see.

Vikash Ranjan · Answer

We have to find the indignation off. It&#x27;s endure. So the bus infection off it is given us when a bomb window blessed when this is Yeah. Relax. Yes. So, assaulting the attend They live equals one. And he was minus tool. Yeah, sort of. Division comes when I&#x27;ve been X minus two about excuse us one Debs clothes. That enough? Minus two. I do have them express who won. See this equal. Do a line X minus. Helen. Text glass. One blessed seen Costanza to them in trouble.

Patrick Vaughn · Answer

this problem were given the inner roll, uh, X squared plus one over X minus one. So we have a numerator with a higher power than the nominator. So instead of using partial fractions, we&#x27;re gonna try using long Division to see if we can get the in a grand and a more manageable form for, But we just want to innovate directly. So in order to do long division, we&#x27;re gonna take X minus one. We&#x27;re going to divide it into X squared plus zero x plus one. I bought the zero X. Here is a placeholder just to make it easier for myself. Eso The first thing we&#x27;re gonna do is make this look like this, have to multiply it by an X we x squared minus X and subtract. That will have expose one. And so next one on what I want. So I have X minus one and was cracked out in. We get a remainder two So, plus two over X minus one. So now our Inter Graham becomes explosive. One plus two over X minus one deals. We can just integrate all these pieces directly. So we get one half X squared plus X plus two Ln absolute value of Xbox one constant and we&#x27;re done

Alex Mangiapane · Answer

So let&#x27;s try out this in ago. Starting off with the U substitution. Let&#x27;s try. U equals two X So d&#x27;you it&#x27;s gonna be to d X. I&#x27;m sorry. Let&#x27;s not make you equal to two X. Let&#x27;s make you eat equal to X squared. We&#x27;re trying to get rid of that ugly X squared. So d&#x27;you is gonna be to X d s. So then this integral is going to become So we know D X can be replaced with the you over to X do you over two acts times one over. Ah, we have this You plus one term times that we still have this X So here&#x27;s another X squared. So this is gonna be the integral of do you. We can pull out the half of you over you plus one, which is perfect for partial fractions. So this is gonna equal a over you plus be over you plus one. So a you plus a plus bu is gonna equal to one, which means that a plus B zero, since they&#x27;re no, you terms over here and a is one these negative one. And just to check that if we plug in a quiz one, we get you plus one minus you. So we just get one. So that all checks out. So we really have the integral of 1/2 won over you, minus one over you. Post one equals 1/2 island. You minus 1/2 Ellen, you plus one plus c. Um, And now we just have to replace you with expert. So 1/2. Ah, Ellen X squared minus Ellen X squared plus one, uh, plus C and that will give us our answer.

Problem

Evaluate the integral. $$\int_{0}^{1} \frac{2}{2…

Problem 10 Easy Difficulty

Evaluate the integral.
$$\int \frac{1}{(x+a)(x+b)} d x$$

Answer

$\frac{1}{b-a} \ln \left|\frac{x+a}{x+b}\right|+C$

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James Stewart

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Anna Marie V.

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Integration Review - Intro

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$\frac{1}{b-a} \ln \left|\frac{x+a}{x+b}\right|+C$

Biocalculus Calculus for the Life Sciences

Anna Marie V.

Heather Z.

Kristen K.

Samuel H.

Integration Review - Intro

Definite vs Indefinite

James StewartBiocalculus Calculus for the Life Sciences

Anna Marie V.

Heather Z.

Kristen K.

Samuel H.

James Stewart

Biocalculus Calculus for the Life Sciences