dx-x-2-x-3-1-1-3-

Question Number 104388 by Ar Brandon last updated on 21/Jul/20

\int \frac{d x}{x^{2} \sqrt[3]{x^{3} - 1}}

Answered by Dwaipayan Shikari last updated on 21/Jul/20

\int \frac{dx}{x^{3} {(1 - \frac{1}{x^{3}})}^{\frac{1}{3}}} = \int \frac{\frac{1}{x^{4}} . x}{{(1 - \frac{1}{x^{3}})}^{\frac{1}{3}}} dx = \frac{x}{3} \int \frac{\frac{3}{x^{4}}}{{(1 - \frac{1}{x^{3}})}^{\frac{1}{3}}} - \frac{1}{3} \int \int \frac{\frac{3}{x^{4}}}{{(1 - \frac{1}{x^{3}})}^{\frac{1}{3}}}

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

= \frac{x}{2} {(1 - \frac{1}{x^{3}})}^{\frac{2}{3}} - \frac{1}{2} \int \frac{3}{x^{4}} . \frac{x^{4}}{3} {(1 - \frac{1}{x^{3}})}^{\frac{2}{3}} dx

\frac{x}{2} {(1 - \frac{1}{x^{3}})}^{\frac{2}{3}} - \frac{1}{6} \int x^{4} t^{\frac{2}{3}} dt {1 - \frac{1}{x^{3}} = t x^{3} = \frac{1}{1 - t}

\frac{x}{2} (1 - \frac{1}{x^{3}}) - \frac{1}{6} \int {(\frac{1}{1 - t})}^{\frac{4}{3}} t^{\frac{2}{3}} \dots . . continue