Aufgabe.

Berechne folgende Integrale.

(1)
$ \mbox{$\int e^x\cos(2x)\,{\mbox{d}}x$}$.
(2)
$ \mbox{$\displaystyle\int {\displaystyle\frac{1 + \cos x}{2 + \sin x}}\,{\mbox{d}}x$}$.
(3)
Seien $ \mbox{$m,\, n\,\in\,\mathbb{N}$}$ . Zu berechnen sind $ \mbox{$\int_0^{2\pi}\cos(mx)\cos(nx)\,{\mbox{d}}x$}$, $ \mbox{$\int_0^{2\pi}\sin(mx)\sin(nx)\,{\mbox{d}}x$}$, $ \mbox{$\int_0^{2\pi}\sin(mx)\cos(nx)\,{\mbox{d}}x$}$.
(4)
$ \mbox{$\displaystyle\int _1^{27}{\displaystyle\frac{{\mbox{d}}x}{\sqrt{x}(1 + \sqrt[3]{x})}}$}$.
(5)
$ \mbox{$\displaystyle\int x^2\sqrt{x^2 + 4}\,{\mbox{d}}x$}$.