Question

Evaluate ${\int }^{}\frac{dx}{x^{\frac{3}{2}}\left(1+\frac{1}{\sqrt{x}}\right)}$

Answer 1

Let $I=\int \frac{dx}{x^{\frac{3}{2}}(1+x^{-\frac{1}{2}})}$

Let $t=1+x^{-\frac{1}{2}}\Rightarrow dt=-\frac{1}{2}{x}^{-\frac{1}{2}-1}dx$

$dt=-\frac{1}{2}{x}^{-\frac{3}{2}}dx=-\frac{1}{2}\frac{dx}{x^{\frac{3}{2}}}$

$\Rightarrow I=\int \frac{-2dt}{t}$

$=-2\log \left|t\right|+c$ where $c$ is the constant of integration.

$=-2\log |1+x^{-\frac{1}{2}}|+c$

$=-2\log |1+\frac{1}{\sqrt{x}}|+c$