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The change of variable used in the previous example

$$2\tan^{-1}t =x$$
$$\dfrac{2}{1+t^2}\,\mathrm{d}t = \mathrm{d}x$$

may seem rather strange, but is indeed very useful. The ultimate reason for this choice is that all trigonometric functions are transformed into rational functions (polynomial quotients)

$$\tan x=\dfrac{2t}{1-t^2}$$
$$\sin x=\dfrac{2t}{1+t^2}$$
$$\cos x=\dfrac{1-t^2}{1+t^2}$$

and there are rules to integrate all rational functions, as we will see later; therefore integrals involving trigonometric functions and additions, substractions, products and quotients may always be integrated with this change of variable. The simpler change of variable

$$t =\tan x$$
$$\tan^{-1}t =x$$
$$\dfrac{1}{1+t^2}\,\mathrm{d}t = \mathrm{d}x$$

would not work here, since in this case $\sin x$ and $\cos x$ involve square roots

$$\tan x=t$$
$$\sin x=\dfrac{t}{\sqrt{1+t^2}}$$
$$\cos x=\dfrac{1}{\sqrt{1+t^2}}$$

However, sometimes other techniques may be shorter (especially for powers of $\sin$ and $\cos$). For instance

$$ \int \sin^2 x\cos x\,\mathrm{d}x \,\overset{\substack{t=\tan\left(\frac{x}{2}\right)\\ 2\tan^{-1}t =x \\ \frac{2}{1+t^2}\,\mathrm{d}t = \mathrm{d}x \\ \phantom{\downarrow}}}{=}\, \int \left(\dfrac{2t}{1+t^2}\right)^2\dfrac{1-t^2}{1+t^2}\dfrac{2}{1+t^2}\,\mathrm{d}t= 8\int \dfrac{t^2(1-t^2)}{(1+t^2)^4}\,\mathrm{d}t $$

leads to a tedious rational function, whereas a wiser change of variable

$$ \int \sin^2 x\cos x\,\mathrm{d}x \,\overset{\substack{t=\sin x\\ \mathrm{d}t=\cos x\,\mathrm{d}x\\ \phantom{\downarrow}}}{=}\, \int t^2\,\mathrm{d}t= \dfrac{1}{3}t^3 + k \,\overset{\substack{t=\sin x\\ \phantom{\downarrow}}}{=}\, \dfrac{1}{3}\sin^3 x + k $$

solves it immediately