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:<math>\begin{align}
\int \cos^2 x \, dx \,&=\, \int \left(\frac{e^{ix}+e^{-ix}}{2}\right)^2 dx \\[6pt]
&=\, \frac14\int \left( e^{2ix}
\end{align}</math>
At this point, it would be possible to change back to real numbers using the formula {{math|''e''<sup>2''ix''</sup> + ''e''<sup>−2''ix''</sup> {{=}} 2 cos 2''x''}}. Alternatively, we can integrate the complex exponentials and not change back to trigonometric functions until the end:
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