Aufgabe.

Berechne die folgenden Integrale.

  1. $ \mbox{$\displaystyle\int_0^{2\pi} \frac{\cos(3t)}{5-4\cos t} \,\text{d}t$}$
  2. $ \mbox{$\displaystyle\int_{-\infty}^\infty \frac{\text{d}x}{(x^2+1)(x^2+4)}$}$
  3. $ \mbox{$\displaystyle\int_{-\infty}^\infty \frac{x^2}{x^4+1}\,\text{d}x$}$
  4. $ \mbox{$\displaystyle\int_{-\infty}^\infty \frac{\sin(\pi x)}{x^3+3x^2+x-5}\,\text{d}x$}$
  5. $ \mbox{$\displaystyle\int_0^\infty \frac{1}{(x^2+1)^n}\,\text{d}x$}$ , wobei $ \mbox{$n\in\mathbb{N}$}$ .