Contoh Soal 8 Materi Integral Tentu Fungsi Aljabar

 $\begin{array}{ll}\\ 36.&\textrm{Jika}\: \: \left [ x \right ]\: \: \textrm{menyatakan bilangan bulat}\\ &\textrm{terbesar yang}\leq x,\: \textrm{maka nilai dari}\\ &\displaystyle \int_{0}^{n^{2}}\left [ \sqrt{x} \right ]\: dx=\: ....\\ &\begin{array}{ll}  \textrm{a}.\quad \displaystyle  \frac{(n+1)n(n+1)}{6}\\ \textrm{b}.\quad \displaystyle \frac{n(n-1)(n-2)}{6}\\ \textrm{c}.\quad \displaystyle \frac{n(n+1)(2n+1)}{6}\\ \textrm{d}.\quad \displaystyle \frac{n(n-1)(3n+1)}{6}\\ \textrm{e}.\quad \color{red}\displaystyle \frac{n(n-1)(4n+1)}{6}\end{array}\\\\ &\textbf{Jawab}:\\  \end{array}$.

$.\: \qquad\begin{aligned}&\displaystyle \int_{0}^{n^{2}}\left [ \sqrt{x} \right ]\: dx\\ &=\displaystyle \int_{0}^{1}\left [\sqrt{0}  \right ]\: dx+\int_{1}^{2^{2}}\left [\sqrt{1}  \right ]\: dx+\int_{2^{2}}^{3^{2}}\left [\sqrt{2^{2}}  \right ]\: dx \\ &\quad +\displaystyle \int_{3^{2}}^{4^{2}}\left [ \sqrt{3^{2}} \right ]\: dx+\cdots +\int_{(n-1)^{2}}^{n^{2}}\left [ \sqrt{(n-1)^{2}} \right ]\: dx\\ &=\displaystyle \int_{0}^{1}0\: dx+\int_{1}^{4}1\: dx+\int_{4}^{9}2\: dx+\int_{9}^{16}3\: dx\\ &\quad +\cdots +\int_{(n-1)^{2}}^{n^{2}}(n-1)\: dx\\ &=0+x|_{1}^{4}+2x|_{4}^{9}+3x|_{9}^{16}+\cdots +(n-1)x|_{(n-1)^{2}}^{n^{2}}\\ &=0+(4-1)+2(9-4)+3(16-9)+\cdots +(n-1)\left ( 2n-1 \right )\\ &=3+2.5+3.7+4.9+5.11+\cdots +(n-1)(2n-1)\\ &=3+10+21+36+55+\cdots +(n-1)(2n-1)\\ &\quad (\textrm{Deretnya berupa barisan aritmetika tingkat 2})\\ &=\displaystyle \frac{n(n-1)(4n+1)}{6} \end{aligned}$.

DAFTAR PUSTAKA

1. Hutahaean. 1980. Kalkulus Diferensial dan Integral I. Jakarta: GRAMEDIA.
2. Kuntarti, Sulistiyono, Kurnianingsih, S. 2007.  Matematika SMA dan MA untuk Kelas XII Semester 1 Program IPA Standar ISI 2006. Jakarta: ESIS.
3. Nugroho, P.A., Gunarto, D. 2013. Big Bank Soal + Bahas Matematika SMA/MA Kelas 1,2,3. Jakarta: WAHYUMEDIA.
4. Sukino. 2015. Matematika Kelompok Peminatan Matematika dan Ilmu Alam. Jakarta: ERLANGGA.
5. Suparmin, S., Malau, A. 2014. Seri Kinomatika: Mainstream Matematika Dasar & Matematika IPA untuk Siswa SMA/MA Kelompok MIA. Bandung: YRAMA WIDYA.

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