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\int \frac{dx}{x\sqrt{1-x^{3}}}

  • Option 1)

    \frac{1}{3}log\left | \frac{\sqrt{1-x^{3}}-1}{\sqrt{1-x^{3}}+1} \right |+c

  • Option 2)

    \frac{1}{3}log\left | \frac{\sqrt{1-x^{3}}+1}{\sqrt{1-x^{3}}-1} \right |+c

  • Option 3)

    \frac{1}{3}log\left | \frac{1}{\sqrt{1-x^{3}}} \right |+c

  • Option 4)

    \frac{1}{3}log\left |1-x^{3} \right |+c

 

Answers (1)

As we learnt

Integration by substitution -

The functions when on substitution of the variable of integration to some quantity gives any one of standard formulas.

 

 

- wherein

Since \int f(x)dx=\int f(t)dt=\int f(\theta )d\theta all variables must be converted into single variable ,\left ( t\, or\ \theta \right )

 

 

 

\int \frac{dx}{x\sqrt{1-x^{3}}}

Put 1- x3 = t2

           \therefore - 3x2 dx = 2tdt

= -\frac{2}{3}\int \frac{dt}{1-t^{2}}=\frac{1}{3}\log \left | \frac{\sqrt{1-x^{3}}-1}{\sqrt{1-x^{3}}+1} \right |+c


Option 1)

\frac{1}{3}log\left | \frac{\sqrt{1-x^{3}}-1}{\sqrt{1-x^{3}}+1} \right |+c

Option 2)

\frac{1}{3}log\left | \frac{\sqrt{1-x^{3}}+1}{\sqrt{1-x^{3}}-1} \right |+c

Option 3)

\frac{1}{3}log\left | \frac{1}{\sqrt{1-x^{3}}} \right |+c

Option 4)

\frac{1}{3}log\left |1-x^{3} \right |+c

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Vakul

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