\ANG{
(a)\quad$\D\int \e^{ax+b}\RMd x$\qquad
(b)\quad$\D\intop_1^{\sqrt{7}}\frac{2x}{x^2-4}\,\RMd x$\\[2mm]
(c)\quad Berechne numerisch (3 Intervalle):\quad$\D\intop_0^{\pi}\sin^2 x\,\RMd x$
} 
\LSG{
\begin{enumerate}
\item $\D\int \e^{ax+b}\RMd x=\frac 1a \e^{ax+b}+C$
\item\mbox{}
\vspace{-8mm}
\begin{align*}
\intop_1^{\sqrt{7}}\frac{2x}{x^2-4}\,\RMd x &=
\intop_1^{\sqrt{7}}\frac{(x^2-4)'}{x^2-4}\,\RMd x=
\lim_{\varepsilon\to 0}\left(
   \left[\ln|x^2-4|\right]_1^{2-\varepsilon}+
     \left[\ln|x^2-4|\right]_{2+\varepsilon}^{\sqrt{7}}\right)=\\
 &=\lim_{\varepsilon\to 0}\left(
\ln|-4\varepsilon+\varepsilon^2|-\ln 3+\ln 3-\ln|4\varepsilon+\varepsilon^2|
\right)=\\
&=\lim_{\varepsilon\to 0}\ln\left|\frac{\varepsilon(-4+\varepsilon)}{\varepsilon(4+\varepsilon)}\right|=
\ln 1=0
\end{align*}
\item $\D\intop_0^{\pi}\sin^2 x\,\RMd x\approx
\frac{\pi}{3}\left(\sin^2\frac{\pi}6+\sin^2\frac{\pi}2+\sin^2\frac{5\pi}6\right)=
\frac{\pi}{3}(\frac 14+1+\frac 14)=\frac{\pi}2
$
\end{enumerate}
}
