# Tagged Questions

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### Computing $\int_0^{2\pi}(1+2\cos t)^n\cos nt\ \mathrm{d}t$

I'd like to calculate the following integral on the interval $[0,2\pi]$: $$I=\int_0^{2\pi}(1+2\cos t)^n\cos nt\ \mathrm{d}t = 2\pi.$$
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### Show these approximations of $\cos$, $\sin$ and $\tan$ are exact.

A while back I was looking for an approximation to $\cos(x)$ and I constructed a polynomial with zeros in the same places as the first few zeros of $cos(x)$: c_n(x) = \frac{\prod_{i=1}^n ...
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### Use Residue Theorem to evaluate $\ \oint_{C_3 (0)} \frac{z+7}{z^4 + z^3 - 2 z^2}\,dz \$?

How do I use Residue Theorem to evaluate $\ \oint_{C_3 (0)} \frac{z+7}{z^4 + z^3 - 2 z^2}\,dz \$ where $C_3(0)$ is the circle of radius 3 centered at the origin, oriented in the counter- clockwise ...
### Singularities of $\ \frac{z-1}{z^2 \sin z} \$
Find all singularities of $\ \frac{z-1}{z^2 \sin z} \$ Determine if they are isolated or nonisolated. This is not hard, it is z = 0 and z = k*pi. But how do I: For isolated singularities, ...
The sum of the squares of the reciprocals of the positive fixed points of the tangent function is $1/10$. I've seen this proved by means of residues, but I don't remember the details. I've also ...