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In this video, the prof has used $x=a\tan u$ so $x=2\tan u$ for the integral of $\cfrac{dx}{(x^2+4)^3}$.

But I thought that could only be used with square roots of this case as in $\displaystyle \int\cfrac{dx}{\sqrt{x^2+4}}$

And without the square root then aren't you dealing with arctan?

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    $\begingroup$ You can use any substitution you like, and if it helps you solve the integral, then so the better ! For example, in your case $x=2\tan u$ will be useful because when you square it you'll get $4\tan^2 u$ which means you can pull out a factor of $\frac{1}{4^3}$ and the integrand becomes $1/(1+\tan^2u)^3=1/\sec^6u=\cos^6u$. Bingo ! $\endgroup$ – Pixel Jul 18 at 17:39
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You can apply any substitution to any integral; you may have seen this particular one recommended for integrals of the form you mention, but that doesn't mean you can't use it elsewhere as well.

Of course, to use a substitution, you need to satisfy the hypotheses of whatever theorem you have that allows substitutions, but in practice, we often proceed by ignoring those, getting a result, and then checking that the derivative works out as we hoped (rather than losing a sign, etc.)

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