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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.

Answer and Explanation: 1


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}


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Evaluating Definite Integrals Using the Fundamental Theorem

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Chapter 16 / Lesson 2
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In calculus, the fundamental theorem is an essential tool that helps explain the relationship between integration and differentiation. Learn about evaluating definite integrals using the fundamental theorem, and work examples to gain understanding.


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