Integration using Euler's formula: Difference between revisions

Revision as of 23:12, 9 October 2017

Revision as of 03:39, 6 March 2018

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:<math>\begin{align}

\int \sin^2 x \cos 4x \, dx

&= \int \left(\frac{e^{ix}-e^{-ix}}{2i}\right)^2\left(\frac{e^{4ix}+e^{-4ix}}{2}\right) dx \\[6pt]

&= -\frac18\int \left(e^{2ix} - 2 + e^{-2ix}\right)\left(e^{4ix}+e^{-4ix}\right) dx \\[6pt]

&= -\frac18\int \left(e^{6ix} - 2e^{4ix} + e^{2ix} + e^{-2ix} - 2e^{-4ix} + e^{-6ix}\right) dx.