Příklad 4.59
[!example] Vypočtěte $\(\Large \int_{0}^{1} \frac{1}{\sqrt{1-x^2}\sqrt[3]{\arccos(x)}}\;dx\)$
\[\large
\begin{align}
&= -\int_{0}^{1} -\frac{1}{\sqrt{1-x^2}}\cdot\frac{1}{\sqrt[3]{\arccos(x)}}\;dx \\
\end{align}
\]
[!tip]+ Substituce
\[\large\begin{aligned} u &= \arccos(x) \\ du &= -\frac{1}{\sqrt{1-x^2}}dx \\ \end{aligned}\]
\[\large
\begin{align}
&= -\int_{\frac{\pi}{2}}^{0} \frac{1}{\sqrt[3]{u}}\;du \\
\end{align}
\]
[!warning] Pozor na prohození integračních mezí viz [[Integrační meze#Otočení integračních mezí]]
\[\large
\begin{align}
&= \int_{0}^{\frac{\pi}{2}} \frac{1}{\sqrt[3]{u}}\;du \\
&= \int_{0}^{\frac{\pi}{2}} u^{-\frac{1}{3}}\;du \\
&= \left[ \frac{u^{\frac{2}{3}}}{\frac{2}{3}}\right]_{0}^{\frac{\pi}{2}} \\
&= \left[ \frac{3}{2}\sqrt[3]{u^2}\right]_{0}^{\frac{\pi}{2}} \\
&= \left[ \frac{3}{2}\sqrt[3]{\left(\frac{\pi}{2}\right)^2}\right] - \left[ \frac{3}{2}\sqrt[3]{0^2}\right]\\
&= \left[ \frac{3}{2}\sqrt[3]{\frac{\pi^2}{4}}\right] - 0\\
&= \boxed{\frac{3}{2}\sqrt[3]{\frac{\pi^2}{4}}}
\end{align}
\]