# How do you use substitution to integrate x^2*(x-2)^(1/2)?

Mar 8, 2018

The answer is $= \frac{2}{7} {\left(x - 2\right)}^{\frac{7}{2}} + \frac{8}{5} {\left(x - 2\right)}^{\frac{5}{2}} + \frac{8}{3} {\left(x - 2\right)}^{\frac{3}{2}} + C$

#### Explanation:

We need

$\int {x}^{n} \mathrm{dx} = {x}^{n + 1} / \left(n + 1\right) + C \left(n \ne - 1\right)$

Perform the substitution

$u = x - 2$, $\implies$, $\mathrm{dx} = \mathrm{du}$

Therefore,

$\int {x}^{2} \sqrt{x - 2} \mathrm{dx} = \int {\left(u + 2\right)}^{2} \sqrt{u} \mathrm{du}$

$= \int \left({u}^{2} + 4 u + 4\right) \sqrt{u} \mathrm{du}$

$= \int \left({u}^{\frac{5}{2}} + 4 {u}^{\frac{3}{2}} + 4 {u}^{\frac{1}{2}}\right) \mathrm{du}$

$= \int {u}^{\frac{5}{2}} \mathrm{du} + 4 \int {u}^{\frac{3}{2}} \mathrm{du} + 4 \int {u}^{\frac{1}{2}} \mathrm{du}$

$= {u}^{\frac{7}{2}} / \left(\frac{7}{2}\right) + 4 {u}^{\frac{5}{2}} / \left(\frac{5}{2}\right) + 4 {u}^{\frac{3}{2}} / \left(\frac{3}{2}\right)$

$= \frac{2}{7} {u}^{\frac{7}{2}} + \frac{8}{5} {u}^{\frac{5}{2}} + \frac{8}{3} {u}^{\frac{3}{2}}$

$= \frac{2}{7} {\left(x - 2\right)}^{\frac{7}{2}} + \frac{8}{5} {\left(x - 2\right)}^{\frac{5}{2}} + \frac{8}{3} {\left(x - 2\right)}^{\frac{3}{2}} + C$