How to evaluate $int frac{x^3}{sqrt {x^2+1}}dx$












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Evaluate $$int frac{x^3}{sqrt {x^2+1}}dx$$
Let $u={x^2}+1$. Then $x=sqrt {u-1}$ and $dx=frac{1}{2sqrt{u-1}}du$.



Therefore, $$int frac{(u-1)sqrt{u-1}}{sqrt u}frac{1}{2sqrt {u-1}}du$$
$$=frac{1}{2}int {sqrt u}-{frac {1}{sqrt u}du}$$
$$={frac{2}{3}}{{(x^2+1})}^{3/2}-{frac{1}{2}}{sqrt {x^2+1}}+C$$for some constant $C$










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    Thank your advice. I correct it.
    $endgroup$
    – Maggie
    Dec 20 '18 at 16:38
















1












$begingroup$


Evaluate $$int frac{x^3}{sqrt {x^2+1}}dx$$
Let $u={x^2}+1$. Then $x=sqrt {u-1}$ and $dx=frac{1}{2sqrt{u-1}}du$.



Therefore, $$int frac{(u-1)sqrt{u-1}}{sqrt u}frac{1}{2sqrt {u-1}}du$$
$$=frac{1}{2}int {sqrt u}-{frac {1}{sqrt u}du}$$
$$={frac{2}{3}}{{(x^2+1})}^{3/2}-{frac{1}{2}}{sqrt {x^2+1}}+C$$for some constant $C$










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  • $begingroup$
    Thank your advice. I correct it.
    $endgroup$
    – Maggie
    Dec 20 '18 at 16:38














1












1








1





$begingroup$


Evaluate $$int frac{x^3}{sqrt {x^2+1}}dx$$
Let $u={x^2}+1$. Then $x=sqrt {u-1}$ and $dx=frac{1}{2sqrt{u-1}}du$.



Therefore, $$int frac{(u-1)sqrt{u-1}}{sqrt u}frac{1}{2sqrt {u-1}}du$$
$$=frac{1}{2}int {sqrt u}-{frac {1}{sqrt u}du}$$
$$={frac{2}{3}}{{(x^2+1})}^{3/2}-{frac{1}{2}}{sqrt {x^2+1}}+C$$for some constant $C$










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Evaluate $$int frac{x^3}{sqrt {x^2+1}}dx$$
Let $u={x^2}+1$. Then $x=sqrt {u-1}$ and $dx=frac{1}{2sqrt{u-1}}du$.



Therefore, $$int frac{(u-1)sqrt{u-1}}{sqrt u}frac{1}{2sqrt {u-1}}du$$
$$=frac{1}{2}int {sqrt u}-{frac {1}{sqrt u}du}$$
$$={frac{2}{3}}{{(x^2+1})}^{3/2}-{frac{1}{2}}{sqrt {x^2+1}}+C$$for some constant $C$







integration proof-verification






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edited Dec 20 '18 at 16:48









J.G.

28.7k22845




28.7k22845










asked Dec 20 '18 at 16:30









MaggieMaggie

888




888












  • $begingroup$
    Thank your advice. I correct it.
    $endgroup$
    – Maggie
    Dec 20 '18 at 16:38


















  • $begingroup$
    Thank your advice. I correct it.
    $endgroup$
    – Maggie
    Dec 20 '18 at 16:38
















$begingroup$
Thank your advice. I correct it.
$endgroup$
– Maggie
Dec 20 '18 at 16:38




$begingroup$
Thank your advice. I correct it.
$endgroup$
– Maggie
Dec 20 '18 at 16:38










4 Answers
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You made a small mistake on the last line; integrating $frac{1}{2}u^{1/2}-frac{1}{2}u^{-1/2}$ should give $frac{1}{3}u^{3/2}-u^{1/2}+C$, not $frac{2}{3}u^{3/2}-frac{1}{2}u^{1/2}+C$. When you want to double-check an antiderivative, it's worth differentiating it to see if you get the expected integrand.






share|cite|improve this answer









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  • $begingroup$
    Sorry. I didn't notice it. Thank a lot!
    $endgroup$
    – Maggie
    Dec 20 '18 at 16:42



















2












$begingroup$

Here I give you an alternative approach. If we make $x = sinh(u)$, we get



begin{align*}
intfrac{x^{3}}{sqrt{x^{2}+1}}mathrm{d}x & = intsinh^{3}(u)mathrm{d}u = int(cosh^{2}(u)-1)sinh(u)mathrm{d}u\
& = frac{cosh^{3}(u)}{3} - cosh(u) + K
end{align*}



Since $x = sinh(u) = pmsqrt{cosh^{2}(u)-1}$, we conclude that $cosh(u) = sqrt{x^{2} + 1}$. Finally, it results that



begin{align*}
intfrac{x^{3}}{sqrt{x^{2}+1}}mathrm{d}x = frac{(x^{2}+1)^{3/2}}{3} - sqrt{x^{2}+1} + K
end{align*}






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  • 1




    $begingroup$
    OH, I didn't find this method.It's helpful for me.Thank a lot.
    $endgroup$
    – Maggie
    Dec 20 '18 at 16:49



















2












$begingroup$

Just to give another approach, let $x^2+1=u^2$, in which case $xdx=udu$ and we have



$$int{x^3oversqrt{x^2+1}}dx=int{x^2cdot xdxoversqrt{x^2+1}}=int{(u^2-1)uduover u}=int(u^2-1)du={1over3}u^3-u+C\={1over3}(x^2+1)^{3/2}-(x^2+1)^{1/2}+C$$



The main virtue (if any) of this approach is that it gets rid of the square root symbol for the integration step.






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    1












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    Method 1:



    Another approach using trigonometric substitutions



    begin{equation}
    I =int frac{x^3}{sqrt {x^2+1}}dx
    end{equation}



    Here let $x = tan(theta)$ we arrive at:



    begin{align}
    I &=int frac{tan^3(theta)}{sqrt {tan^2(theta)+1}}sec^2(theta):dtheta = int frac{tan^3(theta)}{sec(theta)}sec^2(theta):dtheta\
    &= int tan^3(theta)sec(theta):dtheta = int sec(theta)tan(theta) tan^2(theta):dtheta \
    &= int sec(theta)tan(theta) left(sec^2(theta) - 1right):dtheta
    end{align}



    Now let $u = sec(theta)$ to yield:



    begin{equation}
    int sec(theta)tan(theta)left(u^2 - 1right) frac{:dtheta}{sec(theta)tan(theta)} = int u^2 - 1 :du = frac{u^3}{3} - u + C
    end{equation}



    Where $C$ is the constant of integration.



    Now $u = sec(theta)$ and $x = tan(theta)$ and so $u = sec(arctan(x)) = sqrt{x^2 + 1}$



    Thus,



    begin{equation}
    I =int frac{x^3}{sqrt {x^2+1}}dx = frac{1}{3}left( sqrt {x^2+1}right)^3 - sqrt {x^2+1} + C = frac{1}{3}left(x^2+1right)^{frac{3}{2}} - sqrt {x^2+1} + C
    end{equation}



    Method 2:



    begin{align}
    I &=int frac{x^3}{sqrt {x^2+1}}dx = int x cdot frac{x^2}{sqrt {x^2+1}}dx \
    &= int x cdot frac{x^2 + 1 - 1}{sqrt {x^2+1}}dx = int x left( sqrt {x^2+1} - frac{1}{sqrt{x^2+1}}right)dx
    end{align}



    Here let $u = x^2 + 1$:



    begin{align}
    I &= int x left( sqrt {u} - frac{1}{sqrt{u}}right)frac{du}{2x} = frac{1}{2}left[int left( u^{frac{1}{2}} - u^{-frac{1}{2}}right)duright] = frac{1}{2}left[frac{ u^{frac{3}{2}}}{frac{3}{2}} + frac{ u^{frac{1}{2}}}{frac{1}{2}} right] + C \
    &= frac{1}{3}u^{frac{3}{2}} + u^{frac{1}{2}} + C = frac{1}{3}left(x^2 + 1right)^{frac{3}{2}} + left(x^2 + 1right)^{frac{1}{2}} + C
    end{align}



    Where $C$ is the constant of integration.






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      4 Answers
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      4 Answers
      4






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      2












      $begingroup$

      You made a small mistake on the last line; integrating $frac{1}{2}u^{1/2}-frac{1}{2}u^{-1/2}$ should give $frac{1}{3}u^{3/2}-u^{1/2}+C$, not $frac{2}{3}u^{3/2}-frac{1}{2}u^{1/2}+C$. When you want to double-check an antiderivative, it's worth differentiating it to see if you get the expected integrand.






      share|cite|improve this answer









      $endgroup$













      • $begingroup$
        Sorry. I didn't notice it. Thank a lot!
        $endgroup$
        – Maggie
        Dec 20 '18 at 16:42
















      2












      $begingroup$

      You made a small mistake on the last line; integrating $frac{1}{2}u^{1/2}-frac{1}{2}u^{-1/2}$ should give $frac{1}{3}u^{3/2}-u^{1/2}+C$, not $frac{2}{3}u^{3/2}-frac{1}{2}u^{1/2}+C$. When you want to double-check an antiderivative, it's worth differentiating it to see if you get the expected integrand.






      share|cite|improve this answer









      $endgroup$













      • $begingroup$
        Sorry. I didn't notice it. Thank a lot!
        $endgroup$
        – Maggie
        Dec 20 '18 at 16:42














      2












      2








      2





      $begingroup$

      You made a small mistake on the last line; integrating $frac{1}{2}u^{1/2}-frac{1}{2}u^{-1/2}$ should give $frac{1}{3}u^{3/2}-u^{1/2}+C$, not $frac{2}{3}u^{3/2}-frac{1}{2}u^{1/2}+C$. When you want to double-check an antiderivative, it's worth differentiating it to see if you get the expected integrand.






      share|cite|improve this answer









      $endgroup$



      You made a small mistake on the last line; integrating $frac{1}{2}u^{1/2}-frac{1}{2}u^{-1/2}$ should give $frac{1}{3}u^{3/2}-u^{1/2}+C$, not $frac{2}{3}u^{3/2}-frac{1}{2}u^{1/2}+C$. When you want to double-check an antiderivative, it's worth differentiating it to see if you get the expected integrand.







      share|cite|improve this answer












      share|cite|improve this answer



      share|cite|improve this answer










      answered Dec 20 '18 at 16:36









      J.G.J.G.

      28.7k22845




      28.7k22845












      • $begingroup$
        Sorry. I didn't notice it. Thank a lot!
        $endgroup$
        – Maggie
        Dec 20 '18 at 16:42


















      • $begingroup$
        Sorry. I didn't notice it. Thank a lot!
        $endgroup$
        – Maggie
        Dec 20 '18 at 16:42
















      $begingroup$
      Sorry. I didn't notice it. Thank a lot!
      $endgroup$
      – Maggie
      Dec 20 '18 at 16:42




      $begingroup$
      Sorry. I didn't notice it. Thank a lot!
      $endgroup$
      – Maggie
      Dec 20 '18 at 16:42











      2












      $begingroup$

      Here I give you an alternative approach. If we make $x = sinh(u)$, we get



      begin{align*}
      intfrac{x^{3}}{sqrt{x^{2}+1}}mathrm{d}x & = intsinh^{3}(u)mathrm{d}u = int(cosh^{2}(u)-1)sinh(u)mathrm{d}u\
      & = frac{cosh^{3}(u)}{3} - cosh(u) + K
      end{align*}



      Since $x = sinh(u) = pmsqrt{cosh^{2}(u)-1}$, we conclude that $cosh(u) = sqrt{x^{2} + 1}$. Finally, it results that



      begin{align*}
      intfrac{x^{3}}{sqrt{x^{2}+1}}mathrm{d}x = frac{(x^{2}+1)^{3/2}}{3} - sqrt{x^{2}+1} + K
      end{align*}






      share|cite|improve this answer











      $endgroup$









      • 1




        $begingroup$
        OH, I didn't find this method.It's helpful for me.Thank a lot.
        $endgroup$
        – Maggie
        Dec 20 '18 at 16:49
















      2












      $begingroup$

      Here I give you an alternative approach. If we make $x = sinh(u)$, we get



      begin{align*}
      intfrac{x^{3}}{sqrt{x^{2}+1}}mathrm{d}x & = intsinh^{3}(u)mathrm{d}u = int(cosh^{2}(u)-1)sinh(u)mathrm{d}u\
      & = frac{cosh^{3}(u)}{3} - cosh(u) + K
      end{align*}



      Since $x = sinh(u) = pmsqrt{cosh^{2}(u)-1}$, we conclude that $cosh(u) = sqrt{x^{2} + 1}$. Finally, it results that



      begin{align*}
      intfrac{x^{3}}{sqrt{x^{2}+1}}mathrm{d}x = frac{(x^{2}+1)^{3/2}}{3} - sqrt{x^{2}+1} + K
      end{align*}






      share|cite|improve this answer











      $endgroup$









      • 1




        $begingroup$
        OH, I didn't find this method.It's helpful for me.Thank a lot.
        $endgroup$
        – Maggie
        Dec 20 '18 at 16:49














      2












      2








      2





      $begingroup$

      Here I give you an alternative approach. If we make $x = sinh(u)$, we get



      begin{align*}
      intfrac{x^{3}}{sqrt{x^{2}+1}}mathrm{d}x & = intsinh^{3}(u)mathrm{d}u = int(cosh^{2}(u)-1)sinh(u)mathrm{d}u\
      & = frac{cosh^{3}(u)}{3} - cosh(u) + K
      end{align*}



      Since $x = sinh(u) = pmsqrt{cosh^{2}(u)-1}$, we conclude that $cosh(u) = sqrt{x^{2} + 1}$. Finally, it results that



      begin{align*}
      intfrac{x^{3}}{sqrt{x^{2}+1}}mathrm{d}x = frac{(x^{2}+1)^{3/2}}{3} - sqrt{x^{2}+1} + K
      end{align*}






      share|cite|improve this answer











      $endgroup$



      Here I give you an alternative approach. If we make $x = sinh(u)$, we get



      begin{align*}
      intfrac{x^{3}}{sqrt{x^{2}+1}}mathrm{d}x & = intsinh^{3}(u)mathrm{d}u = int(cosh^{2}(u)-1)sinh(u)mathrm{d}u\
      & = frac{cosh^{3}(u)}{3} - cosh(u) + K
      end{align*}



      Since $x = sinh(u) = pmsqrt{cosh^{2}(u)-1}$, we conclude that $cosh(u) = sqrt{x^{2} + 1}$. Finally, it results that



      begin{align*}
      intfrac{x^{3}}{sqrt{x^{2}+1}}mathrm{d}x = frac{(x^{2}+1)^{3/2}}{3} - sqrt{x^{2}+1} + K
      end{align*}







      share|cite|improve this answer














      share|cite|improve this answer



      share|cite|improve this answer








      edited Dec 20 '18 at 16:54

























      answered Dec 20 '18 at 16:40









      APC89APC89

      2,456520




      2,456520








      • 1




        $begingroup$
        OH, I didn't find this method.It's helpful for me.Thank a lot.
        $endgroup$
        – Maggie
        Dec 20 '18 at 16:49














      • 1




        $begingroup$
        OH, I didn't find this method.It's helpful for me.Thank a lot.
        $endgroup$
        – Maggie
        Dec 20 '18 at 16:49








      1




      1




      $begingroup$
      OH, I didn't find this method.It's helpful for me.Thank a lot.
      $endgroup$
      – Maggie
      Dec 20 '18 at 16:49




      $begingroup$
      OH, I didn't find this method.It's helpful for me.Thank a lot.
      $endgroup$
      – Maggie
      Dec 20 '18 at 16:49











      2












      $begingroup$

      Just to give another approach, let $x^2+1=u^2$, in which case $xdx=udu$ and we have



      $$int{x^3oversqrt{x^2+1}}dx=int{x^2cdot xdxoversqrt{x^2+1}}=int{(u^2-1)uduover u}=int(u^2-1)du={1over3}u^3-u+C\={1over3}(x^2+1)^{3/2}-(x^2+1)^{1/2}+C$$



      The main virtue (if any) of this approach is that it gets rid of the square root symbol for the integration step.






      share|cite|improve this answer









      $endgroup$


















        2












        $begingroup$

        Just to give another approach, let $x^2+1=u^2$, in which case $xdx=udu$ and we have



        $$int{x^3oversqrt{x^2+1}}dx=int{x^2cdot xdxoversqrt{x^2+1}}=int{(u^2-1)uduover u}=int(u^2-1)du={1over3}u^3-u+C\={1over3}(x^2+1)^{3/2}-(x^2+1)^{1/2}+C$$



        The main virtue (if any) of this approach is that it gets rid of the square root symbol for the integration step.






        share|cite|improve this answer









        $endgroup$
















          2












          2








          2





          $begingroup$

          Just to give another approach, let $x^2+1=u^2$, in which case $xdx=udu$ and we have



          $$int{x^3oversqrt{x^2+1}}dx=int{x^2cdot xdxoversqrt{x^2+1}}=int{(u^2-1)uduover u}=int(u^2-1)du={1over3}u^3-u+C\={1over3}(x^2+1)^{3/2}-(x^2+1)^{1/2}+C$$



          The main virtue (if any) of this approach is that it gets rid of the square root symbol for the integration step.






          share|cite|improve this answer









          $endgroup$



          Just to give another approach, let $x^2+1=u^2$, in which case $xdx=udu$ and we have



          $$int{x^3oversqrt{x^2+1}}dx=int{x^2cdot xdxoversqrt{x^2+1}}=int{(u^2-1)uduover u}=int(u^2-1)du={1over3}u^3-u+C\={1over3}(x^2+1)^{3/2}-(x^2+1)^{1/2}+C$$



          The main virtue (if any) of this approach is that it gets rid of the square root symbol for the integration step.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 20 '18 at 17:06









          Barry CipraBarry Cipra

          59.9k654126




          59.9k654126























              1












              $begingroup$

              Method 1:



              Another approach using trigonometric substitutions



              begin{equation}
              I =int frac{x^3}{sqrt {x^2+1}}dx
              end{equation}



              Here let $x = tan(theta)$ we arrive at:



              begin{align}
              I &=int frac{tan^3(theta)}{sqrt {tan^2(theta)+1}}sec^2(theta):dtheta = int frac{tan^3(theta)}{sec(theta)}sec^2(theta):dtheta\
              &= int tan^3(theta)sec(theta):dtheta = int sec(theta)tan(theta) tan^2(theta):dtheta \
              &= int sec(theta)tan(theta) left(sec^2(theta) - 1right):dtheta
              end{align}



              Now let $u = sec(theta)$ to yield:



              begin{equation}
              int sec(theta)tan(theta)left(u^2 - 1right) frac{:dtheta}{sec(theta)tan(theta)} = int u^2 - 1 :du = frac{u^3}{3} - u + C
              end{equation}



              Where $C$ is the constant of integration.



              Now $u = sec(theta)$ and $x = tan(theta)$ and so $u = sec(arctan(x)) = sqrt{x^2 + 1}$



              Thus,



              begin{equation}
              I =int frac{x^3}{sqrt {x^2+1}}dx = frac{1}{3}left( sqrt {x^2+1}right)^3 - sqrt {x^2+1} + C = frac{1}{3}left(x^2+1right)^{frac{3}{2}} - sqrt {x^2+1} + C
              end{equation}



              Method 2:



              begin{align}
              I &=int frac{x^3}{sqrt {x^2+1}}dx = int x cdot frac{x^2}{sqrt {x^2+1}}dx \
              &= int x cdot frac{x^2 + 1 - 1}{sqrt {x^2+1}}dx = int x left( sqrt {x^2+1} - frac{1}{sqrt{x^2+1}}right)dx
              end{align}



              Here let $u = x^2 + 1$:



              begin{align}
              I &= int x left( sqrt {u} - frac{1}{sqrt{u}}right)frac{du}{2x} = frac{1}{2}left[int left( u^{frac{1}{2}} - u^{-frac{1}{2}}right)duright] = frac{1}{2}left[frac{ u^{frac{3}{2}}}{frac{3}{2}} + frac{ u^{frac{1}{2}}}{frac{1}{2}} right] + C \
              &= frac{1}{3}u^{frac{3}{2}} + u^{frac{1}{2}} + C = frac{1}{3}left(x^2 + 1right)^{frac{3}{2}} + left(x^2 + 1right)^{frac{1}{2}} + C
              end{align}



              Where $C$ is the constant of integration.






              share|cite|improve this answer











              $endgroup$


















                1












                $begingroup$

                Method 1:



                Another approach using trigonometric substitutions



                begin{equation}
                I =int frac{x^3}{sqrt {x^2+1}}dx
                end{equation}



                Here let $x = tan(theta)$ we arrive at:



                begin{align}
                I &=int frac{tan^3(theta)}{sqrt {tan^2(theta)+1}}sec^2(theta):dtheta = int frac{tan^3(theta)}{sec(theta)}sec^2(theta):dtheta\
                &= int tan^3(theta)sec(theta):dtheta = int sec(theta)tan(theta) tan^2(theta):dtheta \
                &= int sec(theta)tan(theta) left(sec^2(theta) - 1right):dtheta
                end{align}



                Now let $u = sec(theta)$ to yield:



                begin{equation}
                int sec(theta)tan(theta)left(u^2 - 1right) frac{:dtheta}{sec(theta)tan(theta)} = int u^2 - 1 :du = frac{u^3}{3} - u + C
                end{equation}



                Where $C$ is the constant of integration.



                Now $u = sec(theta)$ and $x = tan(theta)$ and so $u = sec(arctan(x)) = sqrt{x^2 + 1}$



                Thus,



                begin{equation}
                I =int frac{x^3}{sqrt {x^2+1}}dx = frac{1}{3}left( sqrt {x^2+1}right)^3 - sqrt {x^2+1} + C = frac{1}{3}left(x^2+1right)^{frac{3}{2}} - sqrt {x^2+1} + C
                end{equation}



                Method 2:



                begin{align}
                I &=int frac{x^3}{sqrt {x^2+1}}dx = int x cdot frac{x^2}{sqrt {x^2+1}}dx \
                &= int x cdot frac{x^2 + 1 - 1}{sqrt {x^2+1}}dx = int x left( sqrt {x^2+1} - frac{1}{sqrt{x^2+1}}right)dx
                end{align}



                Here let $u = x^2 + 1$:



                begin{align}
                I &= int x left( sqrt {u} - frac{1}{sqrt{u}}right)frac{du}{2x} = frac{1}{2}left[int left( u^{frac{1}{2}} - u^{-frac{1}{2}}right)duright] = frac{1}{2}left[frac{ u^{frac{3}{2}}}{frac{3}{2}} + frac{ u^{frac{1}{2}}}{frac{1}{2}} right] + C \
                &= frac{1}{3}u^{frac{3}{2}} + u^{frac{1}{2}} + C = frac{1}{3}left(x^2 + 1right)^{frac{3}{2}} + left(x^2 + 1right)^{frac{1}{2}} + C
                end{align}



                Where $C$ is the constant of integration.






                share|cite|improve this answer











                $endgroup$
















                  1












                  1








                  1





                  $begingroup$

                  Method 1:



                  Another approach using trigonometric substitutions



                  begin{equation}
                  I =int frac{x^3}{sqrt {x^2+1}}dx
                  end{equation}



                  Here let $x = tan(theta)$ we arrive at:



                  begin{align}
                  I &=int frac{tan^3(theta)}{sqrt {tan^2(theta)+1}}sec^2(theta):dtheta = int frac{tan^3(theta)}{sec(theta)}sec^2(theta):dtheta\
                  &= int tan^3(theta)sec(theta):dtheta = int sec(theta)tan(theta) tan^2(theta):dtheta \
                  &= int sec(theta)tan(theta) left(sec^2(theta) - 1right):dtheta
                  end{align}



                  Now let $u = sec(theta)$ to yield:



                  begin{equation}
                  int sec(theta)tan(theta)left(u^2 - 1right) frac{:dtheta}{sec(theta)tan(theta)} = int u^2 - 1 :du = frac{u^3}{3} - u + C
                  end{equation}



                  Where $C$ is the constant of integration.



                  Now $u = sec(theta)$ and $x = tan(theta)$ and so $u = sec(arctan(x)) = sqrt{x^2 + 1}$



                  Thus,



                  begin{equation}
                  I =int frac{x^3}{sqrt {x^2+1}}dx = frac{1}{3}left( sqrt {x^2+1}right)^3 - sqrt {x^2+1} + C = frac{1}{3}left(x^2+1right)^{frac{3}{2}} - sqrt {x^2+1} + C
                  end{equation}



                  Method 2:



                  begin{align}
                  I &=int frac{x^3}{sqrt {x^2+1}}dx = int x cdot frac{x^2}{sqrt {x^2+1}}dx \
                  &= int x cdot frac{x^2 + 1 - 1}{sqrt {x^2+1}}dx = int x left( sqrt {x^2+1} - frac{1}{sqrt{x^2+1}}right)dx
                  end{align}



                  Here let $u = x^2 + 1$:



                  begin{align}
                  I &= int x left( sqrt {u} - frac{1}{sqrt{u}}right)frac{du}{2x} = frac{1}{2}left[int left( u^{frac{1}{2}} - u^{-frac{1}{2}}right)duright] = frac{1}{2}left[frac{ u^{frac{3}{2}}}{frac{3}{2}} + frac{ u^{frac{1}{2}}}{frac{1}{2}} right] + C \
                  &= frac{1}{3}u^{frac{3}{2}} + u^{frac{1}{2}} + C = frac{1}{3}left(x^2 + 1right)^{frac{3}{2}} + left(x^2 + 1right)^{frac{1}{2}} + C
                  end{align}



                  Where $C$ is the constant of integration.






                  share|cite|improve this answer











                  $endgroup$



                  Method 1:



                  Another approach using trigonometric substitutions



                  begin{equation}
                  I =int frac{x^3}{sqrt {x^2+1}}dx
                  end{equation}



                  Here let $x = tan(theta)$ we arrive at:



                  begin{align}
                  I &=int frac{tan^3(theta)}{sqrt {tan^2(theta)+1}}sec^2(theta):dtheta = int frac{tan^3(theta)}{sec(theta)}sec^2(theta):dtheta\
                  &= int tan^3(theta)sec(theta):dtheta = int sec(theta)tan(theta) tan^2(theta):dtheta \
                  &= int sec(theta)tan(theta) left(sec^2(theta) - 1right):dtheta
                  end{align}



                  Now let $u = sec(theta)$ to yield:



                  begin{equation}
                  int sec(theta)tan(theta)left(u^2 - 1right) frac{:dtheta}{sec(theta)tan(theta)} = int u^2 - 1 :du = frac{u^3}{3} - u + C
                  end{equation}



                  Where $C$ is the constant of integration.



                  Now $u = sec(theta)$ and $x = tan(theta)$ and so $u = sec(arctan(x)) = sqrt{x^2 + 1}$



                  Thus,



                  begin{equation}
                  I =int frac{x^3}{sqrt {x^2+1}}dx = frac{1}{3}left( sqrt {x^2+1}right)^3 - sqrt {x^2+1} + C = frac{1}{3}left(x^2+1right)^{frac{3}{2}} - sqrt {x^2+1} + C
                  end{equation}



                  Method 2:



                  begin{align}
                  I &=int frac{x^3}{sqrt {x^2+1}}dx = int x cdot frac{x^2}{sqrt {x^2+1}}dx \
                  &= int x cdot frac{x^2 + 1 - 1}{sqrt {x^2+1}}dx = int x left( sqrt {x^2+1} - frac{1}{sqrt{x^2+1}}right)dx
                  end{align}



                  Here let $u = x^2 + 1$:



                  begin{align}
                  I &= int x left( sqrt {u} - frac{1}{sqrt{u}}right)frac{du}{2x} = frac{1}{2}left[int left( u^{frac{1}{2}} - u^{-frac{1}{2}}right)duright] = frac{1}{2}left[frac{ u^{frac{3}{2}}}{frac{3}{2}} + frac{ u^{frac{1}{2}}}{frac{1}{2}} right] + C \
                  &= frac{1}{3}u^{frac{3}{2}} + u^{frac{1}{2}} + C = frac{1}{3}left(x^2 + 1right)^{frac{3}{2}} + left(x^2 + 1right)^{frac{1}{2}} + C
                  end{align}



                  Where $C$ is the constant of integration.







                  share|cite|improve this answer














                  share|cite|improve this answer



                  share|cite|improve this answer








                  edited Dec 21 '18 at 4:46

























                  answered Dec 21 '18 at 4:34









                  DavidGDavidG

                  2,4911726




                  2,4911726






























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