Question

\[\int\frac{1}{x\sqrt{1 + x^3}} \text{ dx}\]

Answer 1

\[\text{We have}, \]
\[I = \int\frac{dx}{x \sqrt{1 + x^3}}\]
\[ = \int\frac{x^2 dx}{x^3 \sqrt{1 + x^3}}\]
\[\text{ putting  x}^3 = t\]
\[ \Rightarrow \text{  3 x}^2 \text{ dx }= dt\]
\[ \Rightarrow x^2 dx = \frac{dt}{3}\]
\[ \therefore I = \frac{1}{3}\int\frac{dt}{t\sqrt{1 + t}}\]
\[\text{ let 1 + t = p}^2 \]
\[ \Rightarrow \text{ dt = 2p dp}\]
\[I = \frac{1}{3}\int\frac{\text{ 2p dp}}{\left( p^2 - 1 \right) \times p}\]
\[ = \frac{2}{3}\int\frac{dp}{p^2 - 1}\]
\[ = \frac{2}{3} \times \frac{1}{2} \text{ log} \left| \frac{p - 1}{p + 1} \right| + C\]
\[ = \frac{1}{3}\text{ log }\left| \frac{p - 1}{p + 1} \right| + C\]
\[ = \frac{1}{3}\text{ log} \left| \frac{\sqrt{1 + t} - 1}{\sqrt{1 + t} + 1} \right| + C\]
\[ = \frac{1}{3}\text{ log } \left| \frac{\sqrt{1 + x^3} - 1}{\sqrt{1 + x^3} + 1} \right| + C\]