Tentukan hasil dari \( \int e^x \sqrt{1+e^x} \ dx = \cdots \ ? \)
Pembahasan:
Gunakan teknik integral substitusi. Misalkan \( u = 1 + e^x \) sehingga diperoleh:
\begin{aligned} u = 1 + e^x \Leftrightarrow \frac{du}{dx} &= e^x \\[8pt] \Leftrightarrow dx &= \frac{du}{e^x} \end{aligned}
Dengan demikian,
\begin{aligned} \int e^x \sqrt{1+e^x} \ dx &= \int e^x (1+e^x)^{1/2} \ dx \\[8pt] &= \int e^x \ u^{1/2} \cdot \frac{du}{e^x} \\[8pt] &= \int u^{1/2} \ du = \frac{2}{3}u^{3/2}+C \\[8pt] &= \frac{2}{3}(1+e^x)^{3/2}+C \\[8pt] &= \frac{2}{3}(1+e^x) \sqrt{1+e^x} + C \end{aligned}