## Wednesday, February 1, 2017

### Neat Little Integral Trick

John D. Cook writes about a useful integration trick by rewriting trigonometric functions as complex variables. He recasts the integral $\int e^{-x} \sin(4x) dx,$ using $e^{ix} = \cos x + i \sin x$ as the imaginary part of $\int e^{-x} e^{4ix} dx.$

The derivation is cleaner (no integration by parts), and you don't have to remember any trig formulae. You can do pretty much any trig integral:\begin{align} \int \cos x dx & = \int e^{ix} dx \\ & = e^{ix}/i \\ & = -i e^{ix}. \end{align} The real part of the last expression is $\sin x$.