# Evaluate the integral: {eq}\displaystyle \int_0^1 x(2^3 \sqrt x + 5^4 \sqrt x) \ dx {/eq}.

## Question:

Evaluate the integral:

{eq}\displaystyle \int_0^1 x(2^3 \sqrt x + 5^4 \sqrt x) \ dx {/eq}.

## Integrating an Algebraic Function:

Consider an algebraic functtion of the form {eq}\displaystyle f(x)=ax^n {/eq}. The integration of such a function is given by,

$$\displaystyle \int ax^n \ dx=a\left[ \frac{x^{n+1}}{n+1}\right]+c$$

where, {eq}\displaystyle a {/eq} can be any real number and {eq}\displaystyle c {/eq} is the constant of integration.

Integration of the given function will be as shown below,

{eq}\displaystyle \begin{align} &\int_{0}^{1} x(2^3 \sqrt x + 5^4 \sqrt x) \ dx\\ =&\int_{0}^{1}[8x^{3/2}+625 x^{3/2} ] \ dx & \because x(\sqrt{x})=x^{3/2}\\ =&\int_{0}^{1}[633x^{3/2} ] \ dx\\ =&633\left[ \frac{x^{5/2}}{5/2}\right]_0^1\\ =&\frac{1266}{5}[(1)-(0)]\\ =&\frac{1266}{5} \end{align} {/eq}