Tentukan hasil dari \( \int e^x (e^x+1)^{ \frac{1}{5} } \ dx = \cdots \ ? \)
Pembahasan:
Gunakan teknik integral substitusi. Misalkan \( u = e^x + 1 \) sehingga diperoleh:
\begin{aligned} u = e^x + 1 \Leftrightarrow \frac{du}{dx} &= e^x \\[8pt] \Leftrightarrow dx &= \frac{du}{e^x} \end{aligned}
Dengan demikian,
\begin{aligned} \int e^x (e^x+1)^{ \frac{1}{5} } \ dx &= \int e^x u^{\frac{1}{5}} \cdot \frac{du}{e^x} \\[8pt] &= \int u^{\frac{1}{5}} \ du = \frac{1}{\frac{1}{5}+1}u^{\frac{1}{5}+1} + C \\[8pt] &= \frac{5}{6}u^{\frac{6}{5}} + C = \frac{5}{6}(e^x+1)^{\frac{6}{5}}+C \end{aligned}