evaluate-the-following-integrals-int-fracd-xx4-1

Question

Evaluate the following integrals.
$$\int \frac{d x}{x^{4}-1}$$

Amit Srivastava · Accepted Answer

Yeah. I Quito the X by export minus one so we can ride this excess choir. Pressman les one minus X squared by exposing minus one p. X. If illiterate in tow, Do part excess square person by ex whole minus one. I just single part. Similarly one minus X square by x four Minus one Energy. It&#x27;s a bar Here we will divide both number time generator by one by of squares shortage one by square under TJX Esquire minus one by excess square BX and in the second case, one minus X squared can be right by organizing negative sign here Excess choir plus one excess carbon minus one and ah, this is finally one plus one by Access Square. Be X edge in this case X minus one Bikes Holy Square less If we live in said, this will come as a numerator form and, if well organized, this it is excess choir minus one DX by excess choir plus one excess choir minus one heritage. It will get cancelled. So this part if we&#x27;ll consider, let that&#x27;s minus one by X equal to t one place one plus one by exes. Go! The X will be dainty. So finally it is duty by the T Square plus T Square. And this is also the simplest form or printing Gerson off Canon Bus X. So I feel replace this fellow as 10 were sex one by two Canon worth I mean by a little too. And the earth is directly turning us x by one plus e now Italy, we continue the value off t as one. Bayrou too standing with us P edge X minus one bikes but to last stand in wealth X by one Bless.

Amit Srivastava · Answer

court full of X dx society into part Gardez. Good X and gorgeous mint X dx. We can plan for gardist X edge well secret squared X minus the X So now separate distant part Courtis critics X dx has this. Yeah, gorgeous. It&#x27;s DX. I just look, Gore Tex equals D minus 60 years B 50 her technically be minus dp and this part as Kozik bx So final answer will be taken by three by the same minus school six squared X dx less PXE minus sticking right three minus cortex. This is the directive division of clinics. Well, right here less and the excessive x not the value of PSG cortex. My escort X by three less cortex. Bless X. Let&#x27;s see.

Amit Srivastava · Answer

classical rex Upon guard Esquire X DX Let I equal Do This part can be straight edge Kosik Square X and one place Courtis Good X as we know one plus Carter Squares Consequence Critics Surname Great Illustrated straight as will be seen. It&#x27;s So God is God X So separate these two parts consequence squared X upon Carter Squared X again, Corsica Square X and card card will get cancelled through TX. I&#x27;m sure we are comfortable and they will put car tax equal to two. Be put cortex, equal toe be let cortex equal duty So minus Kosik Square. Next year it&#x27;ll be so this part will be mine. STT fun peace. And this is direct in Dickerson off minus court So ultimately minus T minus to plus one by minus one minus cortex. So one by T minus cortex put the value of de aza cortex so one by cortex minus cortex, less

Taylor Shimono · Answer

So for this problem, it&#x27;s going to be much easier if we rewrite this as the integral of X to the negative fourth power DX. And from here we can see that we&#x27;re going to be able to use our power rule that says that the mineral of a number is one over that exploded plus one. I&#x27;m so negative four plus one times x to that number plus one power. And since this is an indeterminate interval, we can&#x27;t forget our plus c on the end. Um, so when we simplify, this is going to be negative. 1/3 times X tv Negative three power plus.

Taylor Shimono · Answer

So for this problem, we can see that the term that refunding the integral of is fairly complex. So that means that we&#x27;re going to want to try to use U substitution here. And generally we use the innermost portion of one of the complex parts of our term when we&#x27;re using U substitution. So what I mean by that is our most complex term is this four to the X power to the negative X squared power. So the interest portion of that is negative X squared. So if I take you to be negative x squared, we find that d U is equal to negative two x t x. And in order to substitute this into our equation, we&#x27;re going to need to divide both sides by negative too. So we get negative. 1/2 to you is equal to X t X, and now we&#x27;re going to be able to substitute this into our equation. So this is going to become the interval of negative 1/2 times four to the you power. Do you? And so we know from our rules of integration that the interval of aid to the you power do you is just equal to one over the natural log of a times 80 you power plus c So when we apply this year are inter goal is going to be negative 1/2 times are a cz This four right here So one over the natural log of four times four to the you power and we know that you is negative expert plus C. And so when we simplify this, we are going to end up with our interval is negative for, um to the negative x squared power over the natural log of 16.

Carson Merrill · Answer

So we&#x27;re gonna use decomposition of fractions to solve this. So we&#x27;re gonna first start by decomposing our fraction. When we do this, we&#x27;ll get a over X minus four plus B over X minus four squared plus C over X minus one. Then we multiply by the denominator, which is X minus four squared times X minus one, and we end up getting one equals a Times X minus four times X minus one plus B times X minus one plus C times X minus four squared then at X equals one we get that one equals by this should be 00 So we&#x27;re just left with C times one minus four squared, which gives us C equals 1/9 and then X equals four. We get a that one equals it&#x27;s going to get a 00 So we&#x27;re just left with the times for minus one, which is going to be end up giving us the B equals one third. And then what? X equals zero. We end up getting that one equals for a minus are one third. That&#x27;s RB value plus 16/9. Because that&#x27;s not 16 times are see value. We&#x27;re gonna end up getting a equals a negative 1/9. So finally, we can plug this back in two are integral, which is going to give us negative. Um, we&#x27;ll take it out of the interest. Will have negative 1/9 of the integral one over X minus four class. Um, one third times the integral of X minus four to the negative, too DX dx, and then plus 1/9 of the integral from one over X minus one DX. So when we do all this, we end up getting as a result negative 1/9 times the natural log of absolute value of explains for plus three over X minus four time or minus the natural log of X minus one plus c Final answer.

Problem

Evaluate the following integrals. $$\int_{1}^{2} …

Problem 16 Medium Difficulty

Evaluate the following integrals.
$$\int \frac{d x}{x^{4}-1}$$

Answer

$\frac{1}{\sqrt{2}} \operatorname{Tan}^{-1} \frac{\left(x-\frac{1}{x}\right)}{\sqrt{2}}+\operatorname{Tan}^{-1} \frac{x}{1}+c$

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$\frac{1}{\sqrt{2}} \operatorname{Tan}^{-1} \frac{\left(x-\frac{1}{x}\right)}{\sqrt{2}}+\operatorname{Tan}^{-1} \frac{x}{1}+c$

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William Briggs, Lyle Cochran, Bernard GilletCalculus Early Transcendentals

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