Integral check. Is partial fractions the only way?












1












$begingroup$


I'm getting a different answer from wolfram and I have no idea where. I have to integrate:



$$int_0^1 frac{xdx}{(2x+1)^3}$$



Is partial fractions the only way?



So evaluating the fraction first:



$$frac{x}{(2x+1)^3} = frac{A}{2x+1} + frac{B}{(2x+1)^2} + frac{C}{(2x+1)^3}$$



$$x = A(2x+1)^2 + B(2x+1) + C$$



$$x = A(4x^2 + 4x + 1) + 2Bx + B + C$$



$$x = 4Ax^2 + 4Ax + A + 2Bx + B + C$$



$$x = x^2(4A) + x(4A+2B) + A + B + C$$



$4A = 0$ and $4A+2B = 1$ and $A + B + C = 0$ so $A = 0$ and $B=frac{1}{2}$ and $C = frac{-1}{2}$



Is the partial fraction part right?



So then I get:



$$int_0^1 frac{xdx}{(2x+1)^3} = int_0^1 frac{dx}{(2(2x+1)^2)} - int_0^1 frac{dx}{2(2x+1)^3}$$



for both, using $u = 2x+1$ and $frac{du}{dx} = 2$ and $du = 2dx$ and $frac{du}{2} = dx$,
$$frac{1}{4} int frac{du}{u^2} - frac{1}{4} int frac{du}{u^3}$$



$$ = [frac{1}{4} - u^{-1} - frac{1}{4} cdot frac{1}{-2} u^{-2} ]_1^3$$



I'm going the route of changing the limits to the new u and since $u = 2x+1$, when $x = 0, u = 1$ and when $x = 1, u = 3$. Is this the right path?



finally I get



$$[frac{-1}{4}u^{-1} + frac{1}{8}u^{-2} ]_1^3$$



I plug in numbers but I get a different answer than wolfram...






integration







share|cite|improve this question











$endgroup$












  • $begingroup$
    I can't see any mistake in your work. What's the answer given by wolfram?
    $endgroup$
    – Thomas Shelby
    2 hours ago










  • $begingroup$
    They have 1/18 as the answer @ThomasShelby
    $endgroup$
    – Jwan622
    40 mins ago
















1












$begingroup$


I'm getting a different answer from wolfram and I have no idea where. I have to integrate:



$$int_0^1 frac{xdx}{(2x+1)^3}$$



Is partial fractions the only way?



So evaluating the fraction first:



$$frac{x}{(2x+1)^3} = frac{A}{2x+1} + frac{B}{(2x+1)^2} + frac{C}{(2x+1)^3}$$



$$x = A(2x+1)^2 + B(2x+1) + C$$



$$x = A(4x^2 + 4x + 1) + 2Bx + B + C$$



$$x = 4Ax^2 + 4Ax + A + 2Bx + B + C$$



$$x = x^2(4A) + x(4A+2B) + A + B + C$$



$4A = 0$ and $4A+2B = 1$ and $A + B + C = 0$ so $A = 0$ and $B=frac{1}{2}$ and $C = frac{-1}{2}$



Is the partial fraction part right?



So then I get:



$$int_0^1 frac{xdx}{(2x+1)^3} = int_0^1 frac{dx}{(2(2x+1)^2)} - int_0^1 frac{dx}{2(2x+1)^3}$$



for both, using $u = 2x+1$ and $frac{du}{dx} = 2$ and $du = 2dx$ and $frac{du}{2} = dx$,
$$frac{1}{4} int frac{du}{u^2} - frac{1}{4} int frac{du}{u^3}$$



$$ = [frac{1}{4} - u^{-1} - frac{1}{4} cdot frac{1}{-2} u^{-2} ]_1^3$$



I'm going the route of changing the limits to the new u and since $u = 2x+1$, when $x = 0, u = 1$ and when $x = 1, u = 3$. Is this the right path?



finally I get



$$[frac{-1}{4}u^{-1} + frac{1}{8}u^{-2} ]_1^3$$



I plug in numbers but I get a different answer than wolfram...






integration







share|cite|improve this question











$endgroup$












  • $begingroup$
    I can't see any mistake in your work. What's the answer given by wolfram?
    $endgroup$
    – Thomas Shelby
    2 hours ago










  • $begingroup$
    They have 1/18 as the answer @ThomasShelby
    $endgroup$
    – Jwan622
    40 mins ago














1












1








1





$begingroup$


I'm getting a different answer from wolfram and I have no idea where. I have to integrate:



$$int_0^1 frac{xdx}{(2x+1)^3}$$



Is partial fractions the only way?



So evaluating the fraction first:



$$frac{x}{(2x+1)^3} = frac{A}{2x+1} + frac{B}{(2x+1)^2} + frac{C}{(2x+1)^3}$$



$$x = A(2x+1)^2 + B(2x+1) + C$$



$$x = A(4x^2 + 4x + 1) + 2Bx + B + C$$



$$x = 4Ax^2 + 4Ax + A + 2Bx + B + C$$



$$x = x^2(4A) + x(4A+2B) + A + B + C$$



$4A = 0$ and $4A+2B = 1$ and $A + B + C = 0$ so $A = 0$ and $B=frac{1}{2}$ and $C = frac{-1}{2}$



Is the partial fraction part right?



So then I get:



$$int_0^1 frac{xdx}{(2x+1)^3} = int_0^1 frac{dx}{(2(2x+1)^2)} - int_0^1 frac{dx}{2(2x+1)^3}$$



for both, using $u = 2x+1$ and $frac{du}{dx} = 2$ and $du = 2dx$ and $frac{du}{2} = dx$,
$$frac{1}{4} int frac{du}{u^2} - frac{1}{4} int frac{du}{u^3}$$



$$ = [frac{1}{4} - u^{-1} - frac{1}{4} cdot frac{1}{-2} u^{-2} ]_1^3$$



I'm going the route of changing the limits to the new u and since $u = 2x+1$, when $x = 0, u = 1$ and when $x = 1, u = 3$. Is this the right path?



finally I get



$$[frac{-1}{4}u^{-1} + frac{1}{8}u^{-2} ]_1^3$$



I plug in numbers but I get a different answer than wolfram...






integration







share|cite|improve this question











$endgroup$




I'm getting a different answer from wolfram and I have no idea where. I have to integrate:



$$int_0^1 frac{xdx}{(2x+1)^3}$$



Is partial fractions the only way?



So evaluating the fraction first:



$$frac{x}{(2x+1)^3} = frac{A}{2x+1} + frac{B}{(2x+1)^2} + frac{C}{(2x+1)^3}$$



$$x = A(2x+1)^2 + B(2x+1) + C$$



$$x = A(4x^2 + 4x + 1) + 2Bx + B + C$$



$$x = 4Ax^2 + 4Ax + A + 2Bx + B + C$$



$$x = x^2(4A) + x(4A+2B) + A + B + C$$



$4A = 0$ and $4A+2B = 1$ and $A + B + C = 0$ so $A = 0$ and $B=frac{1}{2}$ and $C = frac{-1}{2}$



Is the partial fraction part right?



So then I get:



$$int_0^1 frac{xdx}{(2x+1)^3} = int_0^1 frac{dx}{(2(2x+1)^2)} - int_0^1 frac{dx}{2(2x+1)^3}$$



for both, using $u = 2x+1$ and $frac{du}{dx} = 2$ and $du = 2dx$ and $frac{du}{2} = dx$,
$$frac{1}{4} int frac{du}{u^2} - frac{1}{4} int frac{du}{u^3}$$



$$ = [frac{1}{4} - u^{-1} - frac{1}{4} cdot frac{1}{-2} u^{-2} ]_1^3$$



I'm going the route of changing the limits to the new u and since $u = 2x+1$, when $x = 0, u = 1$ and when $x = 1, u = 3$. Is this the right path?



finally I get



$$[frac{-1}{4}u^{-1} + frac{1}{8}u^{-2} ]_1^3$$



I plug in numbers but I get a different answer than wolfram...






integration




integration






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 2 hours ago









Joshua Mundinger

2,5191028




2,5191028










asked 2 hours ago









Jwan622Jwan622

2,13611530




2,13611530












  • $begingroup$
    I can't see any mistake in your work. What's the answer given by wolfram?
    $endgroup$
    – Thomas Shelby
    2 hours ago










  • $begingroup$
    They have 1/18 as the answer @ThomasShelby
    $endgroup$
    – Jwan622
    40 mins ago


















  • $begingroup$
    I can't see any mistake in your work. What's the answer given by wolfram?
    $endgroup$
    – Thomas Shelby
    2 hours ago










  • $begingroup$
    They have 1/18 as the answer @ThomasShelby
    $endgroup$
    – Jwan622
    40 mins ago
















$begingroup$
I can't see any mistake in your work. What's the answer given by wolfram?
$endgroup$
– Thomas Shelby
2 hours ago




$begingroup$
I can't see any mistake in your work. What's the answer given by wolfram?
$endgroup$
– Thomas Shelby
2 hours ago












$begingroup$
They have 1/18 as the answer @ThomasShelby
$endgroup$
– Jwan622
40 mins ago




$begingroup$
They have 1/18 as the answer @ThomasShelby
$endgroup$
– Jwan622
40 mins ago










3 Answers
3






active

oldest

votes


















3












$begingroup$

Hint: Substitute $x=tfrac12 u -tfrac12$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    You beat me to it.
    $endgroup$
    – randomgirl
    2 hours ago










  • $begingroup$
    If I go your way, I wind up with $frac{1}{2} int frac{1}{u^2} - frac{1}{2} int frac{1}{u^3}$ which is different than what I have at some point. I can't see my error.
    $endgroup$
    – Jwan622
    17 mins ago



















2












$begingroup$

A much much easier way to solve it is by using integration by parts.



Hint: Take $u=x$, $dv = frac{dx}{(2x+1)^3}$.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    $u$ but then $dv$?
    $endgroup$
    – manooooh
    2 hours ago










  • $begingroup$
    @manooooh What do you mean? I don't understand.
    $endgroup$
    – Haris Gusic
    2 hours ago










  • $begingroup$
    I am sorry, I understood that you used sub. My apologies.
    $endgroup$
    – manooooh
    2 hours ago



















1












$begingroup$

1) Observe: $$x= frac 12 times left ( (2x + 1) -1right).$$



2) Use this to obtain $$frac{x}{(2x+1)^3} = frac{1}{2} times frac{1}{(2x+1)^2} - frac 12 times frac{1}{(2x+1)^3}.$$



3) Integrate to get



$$ int frac{x}{(2x+1)^3} dx = -frac{1}{4} (2x+1)^{-1} + frac{1}{8} (2x+1)^{-2}.$$






share|cite|improve this answer









$endgroup$













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    3 Answers
    3






    active

    oldest

    votes








    3 Answers
    3






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    3












    $begingroup$

    Hint: Substitute $x=tfrac12 u -tfrac12$






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      You beat me to it.
      $endgroup$
      – randomgirl
      2 hours ago










    • $begingroup$
      If I go your way, I wind up with $frac{1}{2} int frac{1}{u^2} - frac{1}{2} int frac{1}{u^3}$ which is different than what I have at some point. I can't see my error.
      $endgroup$
      – Jwan622
      17 mins ago
















    3












    $begingroup$

    Hint: Substitute $x=tfrac12 u -tfrac12$






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      You beat me to it.
      $endgroup$
      – randomgirl
      2 hours ago










    • $begingroup$
      If I go your way, I wind up with $frac{1}{2} int frac{1}{u^2} - frac{1}{2} int frac{1}{u^3}$ which is different than what I have at some point. I can't see my error.
      $endgroup$
      – Jwan622
      17 mins ago














    3












    3








    3





    $begingroup$

    Hint: Substitute $x=tfrac12 u -tfrac12$






    share|cite|improve this answer









    $endgroup$



    Hint: Substitute $x=tfrac12 u -tfrac12$







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered 2 hours ago









    MPWMPW

    30.3k12157




    30.3k12157












    • $begingroup$
      You beat me to it.
      $endgroup$
      – randomgirl
      2 hours ago










    • $begingroup$
      If I go your way, I wind up with $frac{1}{2} int frac{1}{u^2} - frac{1}{2} int frac{1}{u^3}$ which is different than what I have at some point. I can't see my error.
      $endgroup$
      – Jwan622
      17 mins ago


















    • $begingroup$
      You beat me to it.
      $endgroup$
      – randomgirl
      2 hours ago










    • $begingroup$
      If I go your way, I wind up with $frac{1}{2} int frac{1}{u^2} - frac{1}{2} int frac{1}{u^3}$ which is different than what I have at some point. I can't see my error.
      $endgroup$
      – Jwan622
      17 mins ago
















    $begingroup$
    You beat me to it.
    $endgroup$
    – randomgirl
    2 hours ago




    $begingroup$
    You beat me to it.
    $endgroup$
    – randomgirl
    2 hours ago












    $begingroup$
    If I go your way, I wind up with $frac{1}{2} int frac{1}{u^2} - frac{1}{2} int frac{1}{u^3}$ which is different than what I have at some point. I can't see my error.
    $endgroup$
    – Jwan622
    17 mins ago




    $begingroup$
    If I go your way, I wind up with $frac{1}{2} int frac{1}{u^2} - frac{1}{2} int frac{1}{u^3}$ which is different than what I have at some point. I can't see my error.
    $endgroup$
    – Jwan622
    17 mins ago











    2












    $begingroup$

    A much much easier way to solve it is by using integration by parts.



    Hint: Take $u=x$, $dv = frac{dx}{(2x+1)^3}$.






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      $u$ but then $dv$?
      $endgroup$
      – manooooh
      2 hours ago










    • $begingroup$
      @manooooh What do you mean? I don't understand.
      $endgroup$
      – Haris Gusic
      2 hours ago










    • $begingroup$
      I am sorry, I understood that you used sub. My apologies.
      $endgroup$
      – manooooh
      2 hours ago
















    2












    $begingroup$

    A much much easier way to solve it is by using integration by parts.



    Hint: Take $u=x$, $dv = frac{dx}{(2x+1)^3}$.






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      $u$ but then $dv$?
      $endgroup$
      – manooooh
      2 hours ago










    • $begingroup$
      @manooooh What do you mean? I don't understand.
      $endgroup$
      – Haris Gusic
      2 hours ago










    • $begingroup$
      I am sorry, I understood that you used sub. My apologies.
      $endgroup$
      – manooooh
      2 hours ago














    2












    2








    2





    $begingroup$

    A much much easier way to solve it is by using integration by parts.



    Hint: Take $u=x$, $dv = frac{dx}{(2x+1)^3}$.






    share|cite|improve this answer









    $endgroup$



    A much much easier way to solve it is by using integration by parts.



    Hint: Take $u=x$, $dv = frac{dx}{(2x+1)^3}$.







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered 2 hours ago









    Haris GusicHaris Gusic

    960116




    960116












    • $begingroup$
      $u$ but then $dv$?
      $endgroup$
      – manooooh
      2 hours ago










    • $begingroup$
      @manooooh What do you mean? I don't understand.
      $endgroup$
      – Haris Gusic
      2 hours ago










    • $begingroup$
      I am sorry, I understood that you used sub. My apologies.
      $endgroup$
      – manooooh
      2 hours ago


















    • $begingroup$
      $u$ but then $dv$?
      $endgroup$
      – manooooh
      2 hours ago










    • $begingroup$
      @manooooh What do you mean? I don't understand.
      $endgroup$
      – Haris Gusic
      2 hours ago










    • $begingroup$
      I am sorry, I understood that you used sub. My apologies.
      $endgroup$
      – manooooh
      2 hours ago
















    $begingroup$
    $u$ but then $dv$?
    $endgroup$
    – manooooh
    2 hours ago




    $begingroup$
    $u$ but then $dv$?
    $endgroup$
    – manooooh
    2 hours ago












    $begingroup$
    @manooooh What do you mean? I don't understand.
    $endgroup$
    – Haris Gusic
    2 hours ago




    $begingroup$
    @manooooh What do you mean? I don't understand.
    $endgroup$
    – Haris Gusic
    2 hours ago












    $begingroup$
    I am sorry, I understood that you used sub. My apologies.
    $endgroup$
    – manooooh
    2 hours ago




    $begingroup$
    I am sorry, I understood that you used sub. My apologies.
    $endgroup$
    – manooooh
    2 hours ago











    1












    $begingroup$

    1) Observe: $$x= frac 12 times left ( (2x + 1) -1right).$$



    2) Use this to obtain $$frac{x}{(2x+1)^3} = frac{1}{2} times frac{1}{(2x+1)^2} - frac 12 times frac{1}{(2x+1)^3}.$$



    3) Integrate to get



    $$ int frac{x}{(2x+1)^3} dx = -frac{1}{4} (2x+1)^{-1} + frac{1}{8} (2x+1)^{-2}.$$






    share|cite|improve this answer









    $endgroup$


















      1












      $begingroup$

      1) Observe: $$x= frac 12 times left ( (2x + 1) -1right).$$



      2) Use this to obtain $$frac{x}{(2x+1)^3} = frac{1}{2} times frac{1}{(2x+1)^2} - frac 12 times frac{1}{(2x+1)^3}.$$



      3) Integrate to get



      $$ int frac{x}{(2x+1)^3} dx = -frac{1}{4} (2x+1)^{-1} + frac{1}{8} (2x+1)^{-2}.$$






      share|cite|improve this answer









      $endgroup$
















        1












        1








        1





        $begingroup$

        1) Observe: $$x= frac 12 times left ( (2x + 1) -1right).$$



        2) Use this to obtain $$frac{x}{(2x+1)^3} = frac{1}{2} times frac{1}{(2x+1)^2} - frac 12 times frac{1}{(2x+1)^3}.$$



        3) Integrate to get



        $$ int frac{x}{(2x+1)^3} dx = -frac{1}{4} (2x+1)^{-1} + frac{1}{8} (2x+1)^{-2}.$$






        share|cite|improve this answer









        $endgroup$



        1) Observe: $$x= frac 12 times left ( (2x + 1) -1right).$$



        2) Use this to obtain $$frac{x}{(2x+1)^3} = frac{1}{2} times frac{1}{(2x+1)^2} - frac 12 times frac{1}{(2x+1)^3}.$$



        3) Integrate to get



        $$ int frac{x}{(2x+1)^3} dx = -frac{1}{4} (2x+1)^{-1} + frac{1}{8} (2x+1)^{-2}.$$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 2 hours ago









        FnacoolFnacool

        5,021511




        5,021511






























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            Berlina muro

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            La berlina muro ĉe Bethaniendamm, 1986 La berlina muro aŭ simple la muro [1] , nomita ankaŭ "kontraŭfaŝisma remparo" en propaganda lingvo de la iama Germana Demokratia Respubliko, estis parto de la interngermana limo, kiu dividis de la 13-a de aŭgusto 1961 ĝis la 9-a de novembro 1989 okcidentan Berlinon de la orienta parto de la urbo kaj de la ĉirkaŭa tereno de la GDR. Ĝi estis unu el la plej famaj simboloj de la malvarma milito kaj de la disdivido de Germanio. Pli ol simpla muro, la berlina muro estis fakte tuta defendokonstruaĵo kun du muroj, ena malplena strio, rondirvojo, gardturoj kaj alarmsistemoj. Dum la provo transiri la severe garditan limon al okcidenta Berlino multaj homoj estis mortigitaj. La preciza nombro de viktimoj estas pridisputita kaj ne certa; la nombroj anoncataj varias inter 86 kaj 238 mortintoj. La malplifortiĝo de Sovetunio kaj la politiko de liberigo gvidita de Miĥail Gorbaĉov ebligis al la orientgermanoj faligi la 9-an de novembro 1989 l
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            Berlina aerponto

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            Flugkoridoroj dum la blokado de Berlino La Berlina aerponto estas la provizigo de okcidenta Berlino per aviadiloj fare de la aliancanoj inter la 23-a de junio 1948 kaj la 12-a de majo 1949 dum la blokado de Berlino fare de la sovetia armeo. Enhavo 1 Historio 1.1 La aerkoridoroj 1.2 Unuaj flugoj 1.3 Plibonigo de la aerponto 1.4 Fino de la aerponto 1.5 Rozinbombiloj 1.6 Bilanco 2 Flughavenoj 2.1 Berlino 2.2 Okcidenta Germanio 3 Aviadiloj de la berlina aerponto 4 15-a datreveno de la aerponto 5 Bibliografio 6 Referencoj 7 Eksteraj ligiloj Historio | General Lucius D. Clay, militguberniestro de la usona okupadzono Jam antaŭ la berlina aerponto estiĝis la malgranda aerponto . La estro de la sovetia militadministracio en Germanio, Vasili Daniloviĉ Sokolovski, ordonis la 1-an de aprilo 1948, kiel reago al la konferenco de Londono inter britoj, francoj kaj reprezentantoj de Benelukso pri aligo de Germa
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