Question

Please answer this question​

Answer 1

We are given the Indefinite integral ;

`{:\implies \quad \displaystyle \sf \int (dx)/(x-x^(3))}`

Take x common from denominator ;

`{:\implies \quad \displaystyle \sf \int (dx)/(x(1-x^(2)))}`

Now , Put ;

`{:\implies \quad \displaystyle \sf x^(2)=u}`

So that ;

`{:\implies \quad \displaystyle \sf dx=(du)/(2√(u))\quad and\quad x=√(u)}`

Now , putting the values ;

`{:\implies \quad \displaystyle \sf \int (1)/(√(u)(1-u))* (1)/(2)* (du)/(√(u))}`

Now , as constant can be taken out of the integrand, so now ;

`{:\implies \quad \displaystyle \sf (1)/(2)\int (du)/(u(1-u))}`

Using partial fraction decomposition, Rewrite the integral as ;

`{:\implies \quad \displaystyle \sf (1)/(2)\int \left((1)/(u)+(1)/(1-u)\right)du}`

As Integrals follow distributive property, so breaking the integral into two integrals, and continuing the integration

`{:\implies \quad \displaystyle \sf (1)/(2)\left(\int (1)/(u)du+\int (1)/(1-u)du\right)}`

`{:\implies \quad \displaystyle \sf (1)/(2)\bigg\u+C}`

`:\implies \quad \displaystyle \sf (1)/(2)(log`

`(u)/(1-u)\bigg`

Putting value of u ;

`{:\implies \quad \therefore \quad \underline{\underline\displaystyle \bf \int (dx)/(x-x^(3))=(1)/(2)log\bigg}}`

This is the required answer

Used Concepts :-

• 
  `{\boxed+C}`

• 
  `{\boxed{\bf{log(a)-log(b)=log\left((a)/(b)\right)}}}`