# How do you integrate  (x + 2)^2 / x^(1/3)?

Oct 14, 2016

Expand and integrate term by term.

#### Explanation:

${\left(x + 2\right)}^{2} / {x}^{\frac{1}{3}} = \frac{{x}^{2} + 4 x + 4}{x} ^ \left(\frac{1}{3}\right)$

$= {x}^{2} / {x}^{\frac{1}{3}} + \frac{4 x}{x} ^ \left(\frac{1}{3}\right) + \frac{4}{x} ^ \left(\frac{1}{3}\right)$

$= {x}^{\frac{5}{3}} + 4 {x}^{\frac{2}{3}} + 4 {x}^{- \frac{1}{3}}$

$\int {\left(x + 2\right)}^{2} / {x}^{\frac{1}{3}} \mathrm{dx} = \int \left({x}^{\frac{5}{3}} + 4 {x}^{\frac{2}{3}} + 4 {x}^{- \frac{1}{3}}\right) \mathrm{dx}$

$= \frac{3}{8} {x}^{\frac{8}{3}} + 4 \cdot \frac{3}{5} {x}^{\frac{5}{3}} + 4 \left(\frac{3}{2}\right) {x}^{\frac{2}{3}} + C$

Simplify as desired.