$\begin{array}{ll}\\ 6.&\displaystyle \int x^{3}\sqrt{x}\: dx=\: ....\\ &\begin{array}{ll}\\ \textrm{a}.\quad \color{red}\displaystyle \frac{2}{9}x^{4}\sqrt{x}+C\\ \textrm{b}.\quad \displaystyle \frac{9}{2}x^{4}\sqrt{x}+C\\ \textrm{c}.\quad \displaystyle \frac{1}{9}x^{4}\sqrt{x}+C\\ \textrm{d}.\quad \displaystyle 9x^{4}\sqrt{x}+C\\ \textrm{e}.\quad \displaystyle x^{4}\sqrt{x}+C\end{array}\\\\ &\textbf{Jawab}:\\ &\begin{aligned}&\displaystyle \int x^{3}\sqrt{x}\: dx=\int x^{3}.x^{\frac{1}{2}}\: dx\\ &=\int x^{3\frac{1}{2}}\: dx=\int x^\frac{7}{2}\: dx=\displaystyle \frac{x^{\frac{7}{2}+1}}{\displaystyle \frac{7}{2}+1}+C\\ &=\displaystyle \frac{x^{\frac{9}{2}}}{\displaystyle \frac{9}{2}}+C=\displaystyle \frac{2}{9}x^{4\frac{1}{2}}+C=\displaystyle \frac{2}{9}x^{4}\sqrt{x}+C \end{aligned} \end{array}$.
$\begin{array}{ll}\\ 7.&\displaystyle \int x\sqrt{x\sqrt[3]{x^{2}}}\: dx=\: ....\\ &\begin{array}{ll}\\ \textrm{a}.\quad \displaystyle \frac{17}{6}x^{2}\sqrt{x\sqrt[3]{x^{2}}}+C\\ \textrm{b}.\quad \color{red}\displaystyle \frac{6}{17}x^{2}\sqrt{x\sqrt[3]{x^{2}}}+C\\ \textrm{c}.\quad \displaystyle x^{2}\sqrt{x\sqrt[3]{x^{2}}}+C\\ \textrm{d}.\quad \displaystyle \frac{6}{17}x\sqrt{x\sqrt[3]{x^{2}}}+C\\ \textrm{e}.\quad \displaystyle \frac{1}{2}x\sqrt{x\sqrt[3]{x^{2}}}+C\end{array}\\\\ &\textbf{Jawab}:\\ &\begin{aligned}&\displaystyle \int x\sqrt{x\sqrt[3]{x^{2}}}\: dx\\&=\displaystyle \int x\left ( x.x^{\frac{2}{3}} \right )^{\frac{1}{2}}\: dx\\ &=\displaystyle \int x^{1+\frac{1}{2}+\frac{2}{6}}\: dx\\ &=\displaystyle \int x^{\frac{11}{6}}\: dx=\displaystyle \frac{x^{\frac{11}{6}+1}}{\displaystyle \frac{11}{6}+1}+C\\ &=\displaystyle \frac{x^{\frac{17}{6}}}{\displaystyle \frac{17}{6}}+C=\displaystyle \frac{6}{17}x^{2}.x^{\frac{5}{6}}+C\\ &=\displaystyle \frac{6}{17}x^{2}\left ( x.x^{\frac{2}{3}} \right )^{\frac{1}{2}}+C\\&=\displaystyle \frac{6}{17}x^{2}\sqrt{x\sqrt[3]{x^{2}}}+C \end{aligned} \end{array}$.
$\begin{array}{ll}\\ 8.&\displaystyle \int x^{2}\sqrt{x^{2}\sqrt[3]{x^{3}}}\: dx=\: ....\\&\begin{array}{ll}\\ \textbf{a}.\quad \displaystyle x^{3}\sqrt{x^{2}\sqrt[3]{x^{3}}}+C\\ \textbf{b}.\quad \displaystyle \frac{27}{6}x^{3}\sqrt{x^{2}\sqrt[3]{x^{3}}}+C\\ \textbf{c}.\quad \color{red}\displaystyle \frac{6}{27}x^{3}\sqrt{x^{2}\sqrt[3]{x^{3}}}+C\\ \textbf{d}.\quad \displaystyle \frac{6}{21}x^{2}\sqrt{x^{2}\sqrt[3]{x^{3}}}+C\\ \textbf{e}.\quad \displaystyle \frac{6}{21}x^{3}\sqrt{x^{2}\sqrt[3]{x^{3}}}+C\end{array}\\\\&\textbf{Jawab}:\\&\begin{aligned}&\displaystyle \int x^{2}\sqrt{x^{2}\sqrt[3]{x^{3}}}\: dx=\int x^{2}\left ( x^{2}.x^{1} \right )^{\frac{1}{2}}\: dx\\ &=\int x^{2+\frac{2}{2}+\frac{1}{2}}\: dx=\int x^{\frac{7}{2}}\: dx=\displaystyle \frac{x^{\frac{7}{2}+1}}{\displaystyle \frac{7}{2}+1}+C\\ &=\displaystyle \frac{x^{\frac{9}{2}}}{\displaystyle \frac{9}{2}}+C\\ &=\displaystyle \frac{2}{9}x^{\frac{9}{2}}+C\\ &=\displaystyle \frac{2.3}{9.3}x^{3}.x^{\frac{3}{2}}+C\\ &=\displaystyle \frac{6}{27}x^{3}\left ( x^{3} \right )^{\frac{1}{2}}+C\\ &=\displaystyle \frac{6}{27}x^{3}\sqrt{x^{2}.x}+C\\ &=\displaystyle \frac{6}{27}x^{3}\sqrt{x^{2}\sqrt[3]{x^{3}}}+C\end{aligned} \end{array}$.
$\begin{array}{ll}\\ 9.&\displaystyle \int x^{3}\sqrt{\displaystyle \frac{1}{x}\sqrt[3]{x^{2}}}\: dx=\: ....\\ &\begin{array}{ll}\\ \textrm{a}.\quad \displaystyle \frac{1}{4}x^{4}\sqrt{\displaystyle \frac{1}{x}\sqrt[3]{x^{2}}}+C\\ \textrm{b}.\quad \color{red}\displaystyle \frac{6}{23}x^{4}\sqrt{\displaystyle \frac{1}{x}\sqrt[3]{x^{2}}}+C\\ \textrm{c}.\quad \displaystyle \frac{23}{6}x^{4}\sqrt{\displaystyle \frac{1}{x}\sqrt[3]{x^{2}}}+C\\ \textrm{d}.\quad \displaystyle \frac{23}{6}x^{3}\sqrt{\displaystyle \frac{1}{x}\sqrt[3]{x^{2}}}+C\\ \textrm{e}.\quad \displaystyle \frac{3}{4}x^{3}\sqrt{\displaystyle \frac{1}{x}\sqrt[3]{x^{2}}}+C\end{array}\\\\ &\textbf{Jawab}:\\ &\begin{aligned}&\displaystyle \int x^{3}\sqrt{\frac{1}{x}\sqrt[3]{x^{2}}}\: dx=\int x^{3}\left ( x^{-1}.x^{\frac{2}{3}} \right )^{\frac{1}{2}}\: dx\\ &=\int x^{3-\frac{1}{2}+\frac{1}{3}}\: dx=\int x^{\frac{17}{6}}\: dx\\ &=\displaystyle \frac{x^{\frac{17}{6}+1}}{\displaystyle \frac{17}{6}+1}+C=\displaystyle \frac{x^{\frac{23}{6}}}{\displaystyle \frac{23}{6}}+C\\ &=\displaystyle \frac{6}{23}x^{\frac{24-1}{6}}+C\\ &=\displaystyle \frac{6}{23}x^{4}.x^{-\frac{1}{6}}+C\\ &=\displaystyle \frac{6}{23}x^{4}.\left ( x^{-\frac{1}{3}} \right )^{\frac{1}{2}}+C\\ &=\displaystyle \frac{6}{23}x^{4}\sqrt{x^{-1+\frac{2}{3}}}+C\\ &=\displaystyle \frac{6}{23}x^{4}\sqrt{\frac{1}{x}\sqrt[3]{x^{2}}}+C\end{aligned} \end{array}$.
$\begin{array}{ll}\\ 10.&\displaystyle \int x\sqrt{\displaystyle \frac{1}{x}\sqrt{x\sqrt{\frac{1}{x}}}}\: dx=\: ....\\ &\begin{array}{ll}\\ \textrm{a}.\quad \displaystyle x^{2}\sqrt{\displaystyle \frac{1}{x}\sqrt{x\sqrt{\frac{1}{x}}}}+C\\\\ \textrm{b}.\quad \displaystyle \frac{13}{8}x^{2}\sqrt{\displaystyle \frac{1}{x}\sqrt{x\sqrt{\frac{1}{x}}}}+C\\\\ \textrm{c}.\quad \color{red}\displaystyle \frac{8}{13}x^{2}\sqrt{\displaystyle \frac{1}{x}\sqrt{x\sqrt{\frac{1}{x}}}}+C\\\\ \textrm{d}.\quad \displaystyle \frac{1}{2}x^{2}\sqrt{\displaystyle \frac{1}{x}\sqrt{x\sqrt{\frac{1}{x}}}}+C\\\\ \textrm{e}.\quad \displaystyle \frac{8}{5}x^{2}\sqrt{\displaystyle \frac{1}{x}\sqrt{x\sqrt{\frac{1}{x}}}}+C\end{array}\\\\ &\textbf{Jawab}:\\ &\begin{aligned}&\displaystyle \int x\sqrt{\frac{1}{x}\sqrt{x\sqrt{\frac{1}{x}}}}\: \: dx\\&=\int x\left ( x^{-1}\left ( x\left ( x^{-1} \right )^{\frac{1}{2}} \right )^{\frac{1}{2}} \right )^{\frac{1}{2}} \: dx\\ &=\int x\left ( x^{-\frac{1}{2}}.x^{\frac{1}{4}}.x^{-\frac{1}{8}} \right )\: dx\\ &=\int x^{1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}}\: dx=\int x^{\frac{5}{8}}\: dx\\ &=\displaystyle \frac{x^{\frac{5}{8}+1}}{\displaystyle \frac{5}{8}+1}+C\\ &=\displaystyle \frac{x^{\frac{13}{8}}}{\displaystyle \frac{13}{8}}+C\\ &=\displaystyle \frac{8}{13}x^{\frac{16-3}{8}}+C\\ &=\displaystyle \frac{8}{13}x^{2}.x^{-\frac{3}{8}}+C\\ &=\displaystyle \frac{8}{13}x^{2}\sqrt{x^{-\frac{3}{4}}}+C\\ &=\displaystyle \frac{8}{13}x^{2}\sqrt{x^{-1+\frac{1}{4}}}+C\\ &=\displaystyle \frac{8}{13}x^{2}\sqrt{\frac{1}{x}.\left ( x^{\frac{1}{2}} \right )^{\frac{1}{2}}}+C\\ &=\displaystyle \frac{8}{13}x^{2}\sqrt{\frac{1}{x}\sqrt{x^{1-\frac{1}{2}}}}+C\\ &=\displaystyle \frac{8}{13}x^{2}\sqrt{\frac{1}{x}\sqrt{x\sqrt{\frac{1}{x}}}}+C\end{aligned} \end{array}$
Tidak ada komentar:
Posting Komentar
Informasi