Soal UN 2018 IPA
Hasil dari \( \int x^2 \ (2-x^3)^{\frac{1}{2}} \ dx = \cdots \ ? \)
- \( -\frac{2}{3} (2-x^3)^{\frac{3}{2}} + C \)
- \( -\frac{1}{2} (2-x^3)^{\frac{3}{2}} + C \)
- \( -\frac{2}{9} (2-x^3)^{\frac{3}{2}} + C \)
- \( \frac{2}{9} (2-x^3)^{\frac{3}{2}} + C \)
- \( \frac{2}{3} (2-x^3)^{\frac{3}{2}} + C \)
Pembahasan:
Untuk menyelesaikan soal ini, misalkan \( u = 2-x^3\) sehingga diperoleh:
\begin{aligned} u=2-x^3 \Leftrightarrow \frac{du}{dx} &= -3x^2 \\[8pt] dx &= -\frac{1}{3x^2} \ du \end{aligned}
Substitusikan hasil di atas ke soal integral, diperoleh:
\begin{aligned} \int x^2 \ (2-x^3)^{\frac{1}{2}} \ dx &= \int x^2 \ u^{\frac{1}{2}} \cdot \left( -\frac{1}{3x^2} \right) \ du \\[8pt] &= - \frac{1}{3} \int u^{\frac{1}{2}} \ du \\[8pt] &= -\frac{1}{3} \cdot \frac{1}{\frac{1}{2}+1} u^{\frac{1}{2}+1} + C \\[8pt] &= - \frac{1}{3} \cdot \frac{2}{3} u^{\frac{3}{2}} + C \\[8pt] &= -\frac{2}{9}(2-x^3)^{\frac{3}{2}} + C \end{aligned}
Jawaban C.