Evaluate the following integrals:

Given:

$\int \sqrt{x}\left(x^{3}-\frac{2}{x}\right) d x$

Opening the bracket, we get,

$\Rightarrow \int\left(x^{\frac{1}{2}} \times x^{3}-x^{\frac{1}{2}} \times \frac{2}{x}\right) d x$

$\Rightarrow \int\left(x^{\frac{1}{2}+3}-x^{\frac{1}{2}-1} \times 2\right) d x$

$\Rightarrow \int\left(x^{\frac{7}{2}}-2 x^{-\frac{1}{2}}\right) d x$

By multiplying,

$\Rightarrow \int x^{\frac{7}{2}} d x-2 \int x^{-\frac{1}{2}} d x$

By applying the formula,

$\int x^{n} d x=\frac{x^{n+1}}{n+1}$

$\Rightarrow \frac{x^{\frac{7}{2}+1}}{\frac{7}{2}+1}-2 \frac{x^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}+c$

$\Rightarrow \frac{x^{\frac{9}{2}}}{\frac{9}{2}}-2 \frac{x^{\frac{1}{2}}}{\frac{1}{2}}+c$

$\Rightarrow \frac{2 x^{\frac{9}{2}}}{9}-4 x^{\frac{1}{2}}+c$