Příklad 4.65
[!example] Vypočtěte $\(\Large \int_{0}^{\frac{\sqrt{2}}{2}} \frac{1}{\sqrt{1-x^2}\cdot\arccos(x)}\;dx\)$
[!tip]+ Substituce
\[\large\begin{aligned} u &= \arccos(x) \\ du &= -\frac{1}{\sqrt{1-x^2}}\;dx \\ \end{aligned}\]
\[\large
\begin{align}
&= -\int_{0}^{\frac{\sqrt{2}}{2}} -\frac{1}{\sqrt{1-x^2}}\cdot\frac{1}{\arccos(x)} \;dx \\
&= -\int_{\frac{\pi}{2}}^{\frac{\pi}{4}} \frac{1}{u}\;du \\
\end{align}
\]
[!warning] Pozor na prohození integračních mezí viz [[Integrační meze#Otočení integračních mezí]]
\[\large
\begin{align}
&= \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{1}{u}\;du \\
&= \left[\;\ln|u|\;\right]_{\frac{\pi}{4}}^{\frac{\pi}{2}} \\
&= \left[\ln\left|\frac{\pi}{2}\right| - \ln\left|\frac{\pi}{4}\right| \right] \\
&= \ln\left|\frac{\cancel{\pi}^1}{\cancel{2}^1}\cdot\frac{\cancel{4}^2}{\cancel{\pi}^1}\right| \\
&= \boxed{\ln(2)}
\end{align}
\]