If \int \frac{dx}{x^{3}\left ( 1+x^{6} \right )^{\frac{2}{3}}}=xf(x)(1+x^{6})^{\frac{1}{3}}+C

Where C is a constant of integration, then the function f(x) is equal to:

 

  • Option 1)

    \frac{3}{x^{2}}

  • Option 2)

    -\frac{1}{6x^{3}}

  • Option 3)

    -\frac{1}{2x^{2}}

  • Option 4)

    -\frac{1}{2x^{3}}

 

Answers (1)

\int \frac{dx}{x^{3}\left ( 1+x^{6} \right )^{\frac{2}{3}}}

\Rightarrow I= \int \frac{dx}{x^{3}\left ( 1+x^{6} \right )^{\frac{2}{3}}}= \int \frac{dx}{x^{3}\left ( \frac{1}{x^{6}}+1 \right )^{\frac{2}{3}}}

Put t=\frac{1}{x^{6}}+1\Rightarrow dt=-\frac{6}{x^{7}}dx

\therefore -\int \frac{1}{6}\frac{1}{t^{\frac{2}{3}}}dt = - \frac{1}{6}\times \frac{t^{\frac{2}{3}}}{\frac{1}{3}}+C

\Rightarrow -\frac{\left ( 1+x^{6} \right )^{\frac{1}{3}}}{2x^{2}}+C

Compare with xf(x)(1+x^{6})^{\frac{1}{3}}

f(x)=-\frac{1}{2x^{3}}

Correct option (4)


Option 1)

\frac{3}{x^{2}}

Option 2)

-\frac{1}{6x^{3}}

Option 3)

-\frac{1}{2x^{2}}

Option 4)

-\frac{1}{2x^{3}}

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