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

  • Option 1)

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

  • Option 2)

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

  • Option 3)

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

  • Option 4)

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

 

Answers (1)

best_answer

As we learned,

 

Integration by PARTIAL Fraction -

\int \frac{p(x)}{q(x)}dx  form :

Where     \frac{p(x)}{q(x)}=G(x)+\frac{H(x)}{q(x)}

- wherein

Where H(x) is a Polynomial with degree Less than q(x)

 

 

\int \frac{x\left ( x^{2}-1 \right )dx}{x^{2}+1}

\Rightarrow \: \int \frac{x\left ( x^{2}-1 \right )-2xdx}{x^{2}+1}=\int xdx-\int \frac{2x}{x^{2}+1}dx

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


Option 1)

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

Option 2)

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

Option 3)

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

Option 4)

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

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