We have to calculate:
$\displaystyle \int_{-\pi}^{\pi} \cos(e^{it})dt.$
Is there something more promising one could try instead of a subsitution $u=e^{it}$?
We have to calculate:
$\displaystyle \int_{-\pi}^{\pi} \cos(e^{it})dt.$
Is there something more promising one could try instead of a subsitution $u=e^{it}$?
Actually the Taylor series approach is fairly straightforward too...
$\int_{-\pi}^{\pi} \cos(e^{it}) \,dt = \int_{-\pi}^{\pi} \sum_{n=0}^{\infty} (-1)^n \frac{e^{2int}}{(2n)!} = \sum_{n=0}^{\infty} (-1)^n \int_{-\pi}^{\pi} \frac{e^{2int}}{(2n)!} \, dt = 2 \pi$.
The exchange of summation and integration is justified by uniform convergence on the unit circle. All terms except the constant disappear.