# Evaluate the integral. {eq}\displaystyle \int_0^2 \dfrac {dx} {\sqrt {256 + x^2}} {/eq}

## Question:

Evaluate the integral.

{eq}\displaystyle \int_0^2 \dfrac {dx} {\sqrt {256 + x^2}} {/eq}

## Evaluate Integral:

Concerning the initial integral, we perform the necessary algebraic operations to find an equivalent expression, which can be compared with a known formula. In this way, we can make comparisons of the integral with the formula and obtain the solution by replacing the respective data.

We perform the algebraic operations to find an equivalent integral

{eq}\begin{align*} \int_0^2 \dfrac{dx}{\sqrt{256+x^2}} &=\int_0^2 \dfrac{dx}{\sqrt{16^2+x^2}} \\ &=\int_0^2 \dfrac{dx}{\sqrt{1+\left(\dfrac{x}{16}\right)^2}} \\ &= arcsinh \left( \dfrac{x}{16} \right) \Big |_0^2 \\ &= arcsinh \left( \dfrac{2}{16} \right) - arcsinh \left( \dfrac{0}{16} \right) \\ &= arcsinh \left( \dfrac{1}{8} \right) \\ &\approx 0.1247 \end{align*} {/eq}