Soal UN 2018 IPA
Tentukan \( \int 2x^2 \ (x^3+2)^5 \ dx = \cdots \ ? \)
- \( \frac{1}{18} (x^3+2)^6 + C \)
- \( \frac{1}{9} (x^3+2)^6 + C \)
- \( \frac{1}{6} (x^3+2)^6 + C \)
- \( \frac{1}{3} (x^3+2)^6 + C \)
- \( \frac{2}{3} (x^3+2)^6 + C \)
Pembahasan:
Misalkan \(u = x^3+2 \) sehingga diperoleh:
\begin{aligned} u=x^3+2 \Leftrightarrow \frac{du}{dx} &= 3x^2 \\[8pt] dx &= \frac{1}{3x^2} \ du \end{aligned}
Substitusi hasil di atas ke soal integral, diperoleh:
\begin{aligned} \int 2x^2 \ (x^3+2)^5 \ dx &= \int 2x^2 \ u^5 \ \frac{1}{3x^2} \ du \\[8pt] &= \frac{2}{3} \int u^5 \ du \\[8pt] &= \frac{2}{3} \cdot \frac{1}{6}u^6 + C \\[8pt] &= \frac{1}{9}(x^3+2)^6 + C \end{aligned}
Jawaban B.