Contoh Soal 2 Materi Integral Tentu Fungsi Aljabar

 $\begin{array}{ll}\\ 6.&\textrm{Hasil dari}\: \: \displaystyle \int_{2}^{4}\left ( -x^{2}+6x-8 \right )dx=\: ....\\ &\begin{array}{ll}  \textrm{a}.\quad \displaystyle \frac{38}{3}\\ \textrm{b}.\quad \displaystyle \frac{26}{3}\\ \textrm{c}.\quad \displaystyle \frac{20}{3}\\ \textrm{d}.\quad \displaystyle \frac{16}{3}\\ \textrm{e}.\quad \color{red}\displaystyle \frac{4}{3} \end{array}\\\\&\textbf{Jawab}:\\&\begin{aligned}&\displaystyle \int_{2}^{4}\left ( -x^{2}+6x-8 \right )dx\\ &=-\displaystyle \frac{1}{3}x^{3}+3x^{2}-8x|_{2}^{4}\\ &=\left ( -\displaystyle \frac{1}{3}(4)^{3}+3(4)^{2}-8(4) \right )-\left ( -\displaystyle \frac{1}{3}(2)^{3}+3(2)^{2}-8(2) \right )\\ &=\left (-\displaystyle \frac{64}{3}+48-32 \right )-\left ( -\displaystyle \frac{8}{3}+12-16 \right )\\ &=-\displaystyle \frac{56}{3}+20=-\displaystyle \frac{56+60}{3}=\color{red}\displaystyle \frac{4}{3}  \end{aligned} \end{array}$.

$\begin{array}{ll}\\ 7.&\textrm{Hasil dari}\: \: \displaystyle \int_{0}^{2}x^{2}\left ( x+2 \right )dx=\: ....\\ &\begin{array}{ll}  \textrm{a}.\quad \displaystyle 6\\ \textrm{b}.\quad 6\displaystyle \frac{1}{3}\\ \textrm{c}.\quad 6\displaystyle \frac{2}{3}\\ \textrm{d}.\quad \color{red}9\displaystyle \frac{1}{3}\\ \textrm{e}.\quad \displaystyle 20 \end{array}\\\\&\textbf{Jawab}:\\&\begin{aligned}&\displaystyle \displaystyle \int_{0}^{2}x^{2}\left ( x+2 \right )dx=\displaystyle \int_{0}^{2}x^{3}+2x^{2}\: \: dx\\ &=\displaystyle \frac{1}{4}x^{4}+\frac{2}{3}x^{3}|_{0}^{2}\\ &=\left ( -\displaystyle \frac{1}{4}(2)^{4}+\frac{2}{3}(2)^{3} \right )-\left ( 0 \right )\\ &=4+\displaystyle \frac{16}{3}=\color{red}\displaystyle \frac{4}{3}  \end{aligned} \end{array}$.

$\begin{array}{ll}\\ 8.&\textrm{Nilai dari}\: \: \displaystyle \int_{0}^{2}18x\sqrt{3x^{2}+4}\: \: dx=\: ....\\ &\begin{array}{ll}  \textrm{a}.\quad \displaystyle 4\\ \textrm{b}.\quad 16\\ \textrm{c}.\quad \color{red}112\\ \textrm{d}.\quad 128\\ \textrm{e}.\quad \displaystyle 168 \end{array}\\\\&\textbf{Jawab}:\\&\begin{aligned}&\textrm{Misalkan}\\  &u=3x^{2}+4\Rightarrow du=6x\: dx\\ &\textrm{Selanjutnya}\\ &\displaystyle \int_{0}^{2}18x\sqrt{3x^{2}+4}\: \: dx=\displaystyle \int_{0}^{2}3\sqrt{u}\: \: dx\\ &=\displaystyle \int_{0}^{2}3u^{.^{\frac{1}{2}}}\: \: dx=2u^{.^{\frac{3}{2}}}\: |_{0}^{2}\\ &=2\left ( 3x^{2}+4 \right )^{.^{\frac{3}{2}}}|_{0}^{2}\\ &=2\left ( 3.2^{2}+4 \right )^{.^{\frac{3}{2}}}-(2(0+4)^{.^{\frac{3}{2}}})\\ &=2.(16)^{.^{\frac{3}{2}}}=2.4^{3}-2.2^{3}=128-16\\ &=\color{red}112 \end{aligned} \end{array}$ .

$\begin{array}{ll}\\ 9.&\textrm{Hasil substitusi dari}\: \: u=x+1\: \: \textrm{pada}\\ &\displaystyle \int_{0}^{1}x^{2}\sqrt{x+1}\: dx=\: ....\\ &\begin{array}{ll}  \textrm{a}.\quad \color{red}\displaystyle \int_{0}^{1}(u-1)^{2}\sqrt{u}\: du\\ \textrm{b}.\quad \displaystyle \int_{0}^{1}(u-1)^{2}\sqrt{u}\: du\\ \textrm{c}.\quad \displaystyle \int_{0}^{1}(u-1)^{2}\sqrt{u}\: du\\ \textrm{d}.\quad \displaystyle \int_{0}^{1}(u-1)^{2}\sqrt{u}\: du\\ \textrm{e}.\quad \displaystyle \int_{0}^{1}(u-1)^{2}\sqrt{u}\: du \end{array}\\\\&\textbf{Jawab}:\\&\begin{aligned}&\textrm{Misalkan}\\  &u=x+1\Rightarrow x=u-1\Rightarrow dx=du\\ &\textrm{Selanjutnya}\\ &\displaystyle \int_{0}^{1}(u-1)^{2}\sqrt{u}\: \: du \end{aligned} \end{array}$.

$\begin{array}{ll}\\ 10.&\textrm{Nilai dari}\: \: \displaystyle \int_{0}^{1}5x(1-x)^{6}\: dx=\: ....\\ &\begin{array}{ll}  \textrm{a}.\quad \displaystyle \frac{75}{56}\\ \textrm{b}.\quad \displaystyle \frac{10}{56}\\ \textrm{c}.\quad \color{red}\displaystyle \frac{5}{56}\\ \textrm{d}.\quad -\displaystyle \frac{7}{56}\\ \textrm{e}.\quad -\displaystyle \frac{10}{56}\end{array}\\\\&\textbf{Jawab}:\\&\begin{aligned}&\textrm{Diketahui soal bentuk}\: \textbf{integral parsial}\\  &\textrm{dengan metode}\: \textbf{Tanzalin}\: \textrm{diperoleh}\\ &\begin{array}{|c|c|}\hline \textrm{Diturunkan}&\textrm{Diintegralkan}\\\hline \color{red}5x&(1-x)^{6}\\\hline \color{blue}5&\color{red}-\displaystyle \frac{1}{7}(1-x)^{7}\\\hline 0&\color{blue}\displaystyle \frac{1}{56}(1-x)^{8}\\\hline \end{array}\\ &\textrm{Sehingga}\\ &\displaystyle \int_{0}^{1}5x(1-x)^{6}\: dx\\ &=(5x)\left ( -\displaystyle \frac{1}{7}(1-x)^{7} \right )-(5)\left ( \displaystyle \frac{1}{56}(1-x)^{8} \right )\: |_{0}^{1}\\ &=(0-0)-\left ( 0-\left ( \displaystyle \frac{5}{56} \right ) \right )\\ &=\color{red}\displaystyle \frac{5}{56} \end{aligned} \end{array}$