Evaluate the following integrals:

Let $\mathrm{x}^{\frac{1}{3}}=\mathrm{t}$

So, $\mathrm{dt}=1 / 3 \mathrm{x}^{\frac{1}{2}-1} \mathrm{dx}$

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

$O r, \frac{d x}{x^{\frac{2}{3}}}=3 d t$

$\int \frac{1}{x^{\frac{2}{3}} \sqrt{x^{\frac{2}{3}}-4}} d x=3 \int \frac{d t}{\sqrt{t^{2}-2^{2}}}$

Since, $\int \frac{1}{\sqrt{\left(x^{2}-a^{2}\right)}} d x=\log \left[x+\sqrt{\left(x^{2}-a^{2}\right)}\right]+c$

$=3 \int \frac{\mathrm{dt}}{\sqrt{\mathrm{t}^{2}-2^{2}}}=3 \log \left[\mathrm{t}+\sqrt{\mathrm{t}^{2}-4}\right]+\mathrm{c}$

$=3 \log \left[x^{\frac{1}{3}}+\sqrt{\left(x^{\frac{1}{3}}\right)^{2}-4}\right]+c=3 \log \left[x^{\frac{1}{3}}+\sqrt{x^{\frac{2}{3}}-4}\right]+c$