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asked Apr 27, 2023 23.4k views

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Whitfin · Answer 1 · 2023-05-03T20:51:29+0000

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

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

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

Putting value of u ;

${:\implies \quad \therefore \quad \underline{\underline +C}}$

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