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Step-by-step Solution

Integrate $\sqrt{x^2-4}$ with respect to x

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Answer

$\frac{1}{2}\sqrt{x^2-4}x-2\ln\left|\frac{x+\sqrt{x^2-4}}{2}\right|+C_0$

Step-by-step explanation

Problem to solve:

$\int_{ }^{ }\left(\sqrt{\left(x^2-4\right)}\right)dx$
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Solve the integral $\int\sqrt{x^2-4}dx$ by trigonometric substitution using the substitution

$\begin{matrix}x=2\sec\left(\theta\right) \\ dx=2\sec\left(\theta\right)\tan\left(\theta\right)d\theta\end{matrix}$
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Substituting in the original integral, we get

$\int2\sqrt{4\sec\left(\theta\right)^2-4}\sec\left(\theta\right)\tan\left(\theta\right)d\theta$

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Answer

$\frac{1}{2}\sqrt{x^2-4}x-2\ln\left|\frac{x+\sqrt{x^2-4}}{2}\right|+C_0$