Příklad 4.49
[!example] Vypočtěte $\(\Large \int_{e}^{e^2} \frac{2+\ln{x}}{x\cdot(\ln^2{x} + 4\ln{x} + 4)}\;dx\)$
[!tip]+ Substituce
\[\large\begin{aligned} u &= \ln{x} \\ du &= \frac{1}{x} \\ \end{aligned}\][!info] $\(\ln{(e)} = 1\)$ $\(\ln{(a^b)} = b\cdot\ln{(a)}\)$
\[
\begin{aligned}
&= \int_{e}^{e^2} \frac{2+\ln{x}}{x\cdot(\ln^2{x} + 4\ln{x} + 4)}\;dx \\
&= \int_{1}^{2} \frac{2+u}{u^2 + 4u + 4}\;du \\
&= \int_{1}^{2} \frac{\cancel{u+2}}{\cancel{(u+2)}\cdot(u+2)}\;du \\
&= \int_{1}^{2} \frac{1}{u+2} \;du \\
&= \left[\ln|u+2|\right]_{1}^{2} \\
&= \ln|4| - \ln|3| \\
&= \boxed{\ln\left|\frac{4}{3}\right|}
\end{aligned}
\]
[!info] $\(\ln{a} - \ln{b} = \ln{\frac{a}{b}}\)$ viz [[Logaritmus]]