# Using the properties of integrals, find: int_0^2 x(2-x)^(1/2) dx

## Question:

Using the properties of integrals, find:

{eq}\displaystyle \int_0^2 x(2-x)^{\frac{1}{2}}\ dx {/eq}

## Substitution Method:

If the integrand of an integral is such that one part of it is the derivative of the other part, then it can be solved using the "substitution method". In this method, we assume a part of the integrand as a variable and we find its derivative. Then we convert the given integral completely in terms of this new variable and solve it.

The given integral is:

$$\int_0^2 x(2-x)^{1/2}$$

We will solve it using substitution.

$$\text{Let } 2-x = u \Rightarrow x =2-u\\ -dx = du \\ dx =-du$$

Substitute these values in the given integral:

\begin{align} \int (2-u)u^{1/2}(- du) & = -\int 2u^{1/2} - u^{3/2} du \\ &=- 2 \left( \dfrac{u^{3/2}}{(3/2)} \right)+ \dfrac{u^{5/2}}{(5/2)} & (\because \int x^{a} d x=\frac{x^{a+1}}{a+1}) \\ & =- \dfrac{4}{3} u^{3/2} + \dfrac{2}{5} u^{5/2} \\ & = \left( - \dfrac{4}{3} (2-x)^{3/2} + \dfrac{2}{5} (2-x)^{5/2}\right)_0^2 & \text{(Substituted the value of u back and applied limits)} \\ &= -\dfrac{4}{3} (0-2^{3/2}) + \dfrac{2}{5} (0-2^{5/2}) \\ &= \dfrac{4}{3} (2 \sqrt{2}) - \dfrac{2}{5} (4 \sqrt{2}) \\ &= \dfrac{8 \sqrt{2}}{3}- \dfrac{8 \sqrt{2}}{5} \\ &= \boxed{\mathbf{\frac{16 \sqrt{2}}{15}}} \end{align} 