int frac{dx}{x(x^3+1)} = | vedclass

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Question

(C) Let $I = \int \frac{dx}{x(x^3+1)}$.Multiply the numerator and denominator by $x^2$:$I = \int \frac{x^2 dx}{x^3(x^3+1)}$.Let $x^3 = t$,then $3x^2 d

Vedclass · Accepted Answer

$\log \left(\sqrt[3]{\frac{x^3}{x^3+1}}\right)+c$,where $c$ is the constant of integration

Vedclass · Answer

(C) Let $I = \int \frac{dx}{x(x^3+1)}$.Multiply the numerator and denominator by $x^2$:$I = \int \frac{x^2 dx}{x^3(x^3+1)}$.Let $x^3 = t$,then $3x^2 dx = dt$,or $x^2 dx = \frac{dt}{3}$.Substituting these into the integral:$I = \frac{1}{3} \int \frac{dt}{t(t+1)}$.Using partial fractions,$\frac{1}{t(t+1)} = \frac{1}{t} - \frac{1}{t+1}$.$I = \frac{1}{3} \int \left(\frac{1}{t} - \frac{1}{t+1}\right) dt$.$I = \frac{1}{3} (\log|t| - \log|t+1|) + c$.$I = \frac{1}{3} \log \left|\frac{t}{t+1}\right| + c$.Substituting $t = x^3$ back:$I = \frac{1}{3} \log \left|\frac{x^3}{x^3+1}\right| + c = \log \left|\left(\frac{x^3}{x^3+1}\right)^{1/3}\right| + c = \log \left(\sqrt[3]{\frac{x^3}{x^3+1}}\right) + c$.

$\int \frac{dx}{x(x^3+1)} = $

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