Příklad 4.2

[!example] Vypočtěte $\(\Large \int_0^{1} \frac{x^4}{\left(e^{x^5}\right)^5}\;dx\)$

\[\large \begin{align} &= \int_0^{1} \frac{x^4}{e^{5x^5}}\;dx \\ &= \frac{1}{25}\int_0^{1} \frac{25x^4}{e^{5x^5}}\;dx \end{align} \]

[!tip]- Substituce

\[\large\begin{aligned} u &= 5x^5 \\ du &= 25x^4 \\ \end{aligned}\]
\[\large \begin{align} &= \frac{1}{25}\int_0^{5} \frac{1}{e^{u}}\;du \\ &= \frac{1}{25}\int_0^{5} e^{-u}\;du \end{align} \]

[!tip]- Substituce

\[\large\begin{aligned} v &= -u \\ dv &= -du \\ \end{aligned}\]
\[\large \begin{align} &= -\frac{1}{25}\int_{0}^{-5} e^{v}\;dv \\ &= \frac{1}{25}\int_{-5}^{0} e^{v}\;dv \\ \end{align} \]

[!warning] Pozor na hranice integrace Pro \(a > b\) platí, že \(\int_a^b f(x) = -\int_b^a f(x)\). viz [[Integrační meze#Otočení integračních mezí]]

\[\large \begin{align} &= \frac{1}{25}\left[e^v\right]^{0}_{-5} \\ &= \frac{1}{25}\cdot\left(e^0 - e^{-5}\right) \\ &= \boxed{\frac{1}{25}\cdot\left(1 - e^{-5}\right)} \\ \end{align} \]