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Evaluate the following integral: \int \frac{x}{\sqrt{x^2 4}} \text{d} x

Question:

Evaluate the following integral:

{eq}\int \frac{x}{\sqrt{x^2-4}} \text{d} x {/eq}

Indefinite Integrals:

In addition to utilizing steps such as variable substitution or other integration techniques to calculate the anti-derivative, a constant of integration (added to the anti-derivative) has to be applied for all indefinite integrals since an indefinite integral does not contain any closed bounds on its symbol.

Answer and Explanation:

Given: {eq}\int \frac{x}{\sqrt{x^2-4}} dx {/eq}


To evaluate this integral, the strategy is to apply variable substitution to the expression to simplify calculations for calculating the anti-derivative.


{eq}\begin{align*} \int \frac{x}{\sqrt{x^2-4}} dx &= \int \frac{1}{\sqrt{u}}\cdot \frac{1}{2} du, (\text{ where } u = x^2-4 \Rightarrow du = 2x dx) \text{ [Variable Substitution]} \\ &= \frac{1}{2}\cdot \int \frac{1}{u^{\frac{1}{2}}} du \\ &= \frac{1}{2}\cdot \int u^{-\frac{1}{2}} du \\ &= \frac{1}{2}\cdot \frac{u^{-\frac{1}{2}+1}}{-\frac{1}{2}+1} \\ &= \frac{1}{2}\cdot \frac{u^{\frac{1}{2}}}{\frac{1}{2}} \\ &= \frac{1}{2}\cdot 2\sqrt{u} \\ &= \sqrt{x^2-4}+c \text{ [Substitution of original expression and addition of constant of integration]} \\ \end{align*} {/eq}


Learn more about this topic:

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Indefinite Integrals as Anti Derivatives

from Math 104: Calculus

Chapter 12 / Lesson 11
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