The integral \int \frac{2x^{3}-1}{x^{4}+x}dx is equal to : (Here C is a constant of integration) 


 

  • Option 1)

    \log_{e}\frac{\left | x^{3}+1 \right |}{x^{2}}+C

  • Option 2)

    \log_{e}\frac{\left | x^{3}+1 \right |}{x}+C

  • Option 3)

    \frac{1}{2}\log_{e}\frac{\left | x^{3}+1 \right |}{x^{2}}+C

     

  • Option 4)

    \frac{1}{2}\log_{e}\frac{\left ( x^{3}+1 \right )^{2}}{\left | x^{3} \right |}+C

 

Answers (1)

\int \frac{2x^{3}-1}{x^{4}+x}dx

=\int \frac{2x-\frac{1}{x^{2}}}{x^{2}+\frac{1}{x}}dx

x^{2}+\frac{1}{x}=t

(2x-\frac{1}{x^{2}})dx=dt

=\int \frac{\mathrm{d} t}{\mathrm{t} }=\ln (t)+C 

                 =\ln \left ( x^{2}+\frac{1}{x} \right )+C

                =\ln \left ( \frac{x^{3}+1}{x} \right )+C


Option 1)

\log_{e}\frac{\left | x^{3}+1 \right |}{x^{2}}+C

Option 2)

\log_{e}\frac{\left | x^{3}+1 \right |}{x}+C

Option 3)

\frac{1}{2}\log_{e}\frac{\left | x^{3}+1 \right |}{x^{2}}+C

 

Option 4)

\frac{1}{2}\log_{e}\frac{\left ( x^{3}+1 \right )^{2}}{\left | x^{3} \right |}+C

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