Tentukan hasil dari \( \int 3x^2 \sqrt{2x^3+5} \ dx = \cdots \ ? \)
Pembahasan:
Misalkan \(u = 2x^3 + 5\), sehingga kita peroleh berikut:
\begin{aligned} u = 2x^3+5 \Leftrightarrow \frac{du}{dx} = 6x^2 \\[8pt] \Leftrightarrow dx = \frac{du}{6x^2} \end{aligned}
Dengan demikian, kita peroleh:
\begin{aligned} \int 3x^2 \sqrt{2x^3+5} \ dx &= \int 3x^2 \cdot \sqrt{u} \cdot \frac{du}{6x^2} \\[8pt] &= \frac{3}{6} \int \sqrt{u} \ du = \frac{1}{2} \int u^{1/2} \ du \\[8pt] &= \frac{1}{2} \cdot \frac{1}{\frac{1}{2}+1} u^{\frac{1}{2}+1} + C \\[8pt] &= \frac{1}{2} \cdot \frac{2}{3} u^{3/2} + C \\[8pt] &= \frac{1}{3} (2x^3+5)^{3/2} + C \quad \text{atau} \\[8pt] &= \frac{1}{3} (2x^3+5) \sqrt{2x^3+5} + C \end{aligned}