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Step-by-step Solution

Integral of $\frac{1}{\sqrt{\left(x^2-4\right)^{3}}}$ with respect to x

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Answer

$\frac{-\frac{1}{4}x}{\sqrt{x^2-4}}+C_0$

Step-by-step explanation

Problem to solve:

$\int\left(\frac{1}{\left(X^2-4\right)^{\frac{3}{2}}}\right)dx$
1

Solve the integral $\int\frac{1}{\sqrt{\left(x^2-4\right)^{3}}}dx$ 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

$\int\frac{2\tan\left(\theta\right)\sec\left(\theta\right)}{\sqrt{\left(4\sec\left(\theta\right)^2-4\right)^{3}}}d\theta$

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Answer

$\frac{-\frac{1}{4}x}{\sqrt{x^2-4}}+C_0$