evaluate-the-indefinite-integral-displaystyle-int-fracx1-x4-dx

Question

Evaluate the indefinite integral. 

$ \displaystyle \int \frac{x}{1 + x^4} \, dx $

Frank Lin · Accepted Answer

evaluate this integral home. In this case, it might not be so obvious that say, maybe we can try something like maybe you want to try something. You whose? One plus x force. But that doesn&#x27;t work very well. Because if you do this to you will be by taking the directive for excuse the X which this does not appear your original formula. So this doesn&#x27;t work. So we have to try something else on DH. Maybe a good way to think about it this way. Want to you, Toby? Something Not necessarily exactly x t x but it should be something like sixty ex who is a constant in front. Ah, That way we can simplify our integral. If you think that way, then you you realize that good choice of you should be X squared and we can write a denominator in terms of excrement And two, you should be too ex DX If you observe this your is off basically solves the problem because x t x part will be one over two You and one plus X forthe is one plus you square and this is one over one Prosecute Score is something you know from the integral table. This entire derivative is our attention. You We have a constant one over two in front plus C here, which is replace you. Buy the origin of functioning of the original variable which in this case use X squared.

J Hardin · Answer

for this in a girl. Let&#x27;s go ahead and do a use up. You equals X players so that X equals you too. The one half And then here let&#x27;s differentiate. So use that power rule and then we can rewrite X is you two the one half and then solve this equation here for D s So we have to do you over to use in the one half. Now let&#x27;s rewrite this integral after the use of So here&#x27;s RDX up here. Well, go ahead and replace that with do you over two square on you and then on the bottom we see that there&#x27;s another exterior. So by our earlier work that&#x27;LL just be swear rule you. And then here we see exit of four plus one. So that&#x27;s just X squared, squared plus one and then using the definition of you, that&#x27;s just you. Square plus one. Now let&#x27;s go ahead and combined. These use here that&#x27;LL just become also has pulled off the two we have won over you and then you square plus one. So before we try partial fracture decomposition, we should look at this quadratic here to see whether factors this one does not factor because the discriminated B squared minus four a c for this problem is negative, for which is less than zero. So we can write this in the form. And were you bu plus c overyou squared plus one and then do you and then of course, would have to go ahead and find and B and C So let me go on to the next page. Here we have one half and then is one. So we have one over x. So really here I should stick with you because that was just a variable, that word. And then I have minus and that you overyou squared plus one. So be is minus one cia zero. And then here let&#x27;s go ahead and integrate this and that here we&#x27;re getting the one half and then for this integral here you can go ahead and do it use up so d w over two equals you. Do you so that the one half is coming from this all here outside the integral. And then we get another one half here and then we have into rule and then we just have dw over and over You. So one half natural log, absolute value. You. And then here we see the one over four natural logged ofyou and it&#x27;s an RC and no. Now let&#x27;s go back and write everything in terms of X. So here w you squared plus one. But you was x square, so we could write that as X square square. So that&#x27;s thanks to the four plus one. So we have one half Ellen and that by definition of you, this is just X square minus one over four natural lug. Excellent four plus one at our constant C at the and that&#x27;s a final answer.

Fuzail Shakir · Answer

So the first thing we&#x27;ve been a news we&#x27;re gonna be right to the interval to be interval of acts over one plus x squared square DX We&#x27;re gonna let you is equal to access square toe That 1/2 d&#x27;you is equal to x d x If you apply the substitution and then pull the 1/2 I won&#x27;t we end up with 1/2 times the interval of d&#x27;you divided by one plus you square which is equal to 1/2 times the arc tangent of you plus C back substituting We end up with 1/2 times the arc tangent of X squared plus C.

J Hardin · Answer

Let&#x27;s evaluate the integral of X to the fourth over X minus one. Here, the first thing we noticed is that the numerator has larger than read. So let&#x27;s go out in these long division X minus one going into X to the fourth power we have executed. We&#x27;Ll supply that subtract left over with X cubed so we can add X squared up here we had X cubed minus X squared That&#x27;s over with X squared so we could come back up here at an ex and then we&#x27;ll have X squared minus x subtract left over with X And then we could come back here once more and then add that one in there and then we&#x27;re X minus one Subtract that off and we had a one left over X Does not go into one. So this is our remainder. So we could come and write this as well. We just did the division our quotient was X cubed Ex player accident one and then we have our remainder divided by the question the original divisor X minus one and this is a much easier in the world. In the original, we can use the power rule for the first four terms for the last one, and may or may not help you to use the use up here. If you need to use because of the minus one, go ahead and do the U equals X minus one. So using the power rule and then the natural log, we have extra the fourth over. Four. Execute over three X squared over two X and then plus natural law. Don&#x27;t forget that. Absolute value X minus one inside and closer constancy of integration and there&#x27;s our answer.

Amrita Bhasin · Answer

We know that if you is X squared plus four, then do you is to axe detox, Which means that ax d X is do you divide it by two change? We can pull the constant warning half out. So you have 1/2 times in a row. One over you, do you, which is 1/2 natural walk of you plus c been substituting in you is X squared plus four. We end up with our solution.

Amrita Bhasin · Answer

okay for this problem of first thing you must recall is that we can break up this integral in tow one over one plus X Square detox, plus the integral of acts over one plus x squared DX. And the reason why is because one plus tax everyone pulls X squared DX is equivalent to these two. We can break them up. Given the fundamental dilemma calculus. Now we know that you is one plus x squared. Take the derivative. We now have inverse tangent of axe post 1/2 times do you over you, which gives us inverse tangent of X. That&#x27;s the first part plus 1/2 and now we can substitute natural log of one plus x squared. Remember, this is what our US plus C

Frank Lin · Answer

okay, Given the problem, the first thing you know you could do is we can rewrite the denominator as one plus ax squared squared D backs. Now we know that you is equivalent to x squared, which means that de X is equivalent to one divide by two acts d&#x27;you, which means now we have the integral of one over one plus u squared times one over two acts d&#x27;you, which means factoring out the 1/2. We know we have 1/2 times inverse tangent because this is inverse tangent plus c. And then remember what you is x squared plus C.

Problem

Evaluate the indefinite integral. $ \displayst…

Problem 44 Hard Difficulty

Evaluate the indefinite integral.

$ \displaystyle \int \frac{x}{1 + x^4} \, dx $

Answer

$\frac{1}{2} \tan ^{-1}\left(x^{2}\right)+\mathrm{C}$

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$\frac{1}{2} \tan ^{-1}\left(x^{2}\right)+\mathrm{C}$

Calculus

Catherine R.

Anna Marie V.

Michael J.

Joseph L.

Precalculus Review - Intro

Functions on the Real Line - Overview

James StewartCalculus

Catherine R.

Anna Marie V.

Michael J.

Joseph L.

James Stewart

Calculus