Příklad 4.56
[!example] Vypočtěte $\(\Large \int_{0}^{2} \frac{-4x}{4+x^2}\;dx\)$
\[\large
\begin{align}
&= -2\int_{0}^{2} \frac{2x}{4+x^2}\;dx \\
\end{align}
\]
[!tip]+ Substituce
\[\large\begin{aligned} u &= x^2 \\ du &= 2xdx \\ \end{aligned}\]
\[\large
\begin{align}
&= -2\int_{0}^{4} \frac{1}{u+4}\;du \\
&= -2\left[\ln\left|u+4\right|\right]_{0}^{4} \\
&= -2\left(\ln\left|8\right| - \ln\left|4\right|\right)\\
&= -2\left(\ln\left|\frac{\cancel{8}^2}{\cancel{4}^1}\right|\right)\\
&= \boxed{-2\ln\left|2\right|}\\
\end{align}
\]