Příklad 4.62
[!example] Vypočtěte $\(\Large \int_{0}^{1} \frac{1}{(1+x^2)\cdot\sqrt{\arctan(x)}}\;dx\)$
[!tip]+ Substituce
\[\large\begin{aligned} u &= \arctan{x} \\ du &= \frac{1}{x^2+1}dx \\ \end{aligned}\]
\[\large
\begin{align}
&= \int_{0}^{\frac{\pi}{4}} \frac{1}{\sqrt{u}}\;du \\
&= \int_{0}^{\frac{\pi}{4}} u^{-\frac{1}{2}}\;du \\
&= \left[\frac{u^{\frac{1}{2}}}{\frac{1}{2}}\right]_{0}^{\frac{\pi}{4}} \\
&= \left[2u^{\frac{1}{2}}\right]_{0}^{\frac{\pi}{4}} \\
&= 2\sqrt{\frac{\pi}{4}}\\
&= \cancel{2}^1{\frac{\sqrt{\pi}}{\cancel{2}^1}}\\
&= \boxed{\sqrt{\pi}}\\
\end{align}
\]