積分(仮)

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\(\displaystyle \int_{0}^{1} x dx = \frac{1}{2} \)
\(\displaystyle \int_{2}^{3} x dx = \frac{5}{2} \)
\(\displaystyle \int_{4}^{5} x dx = \frac{9}{2} \)
\(\displaystyle \int_{6}^{7} x dx = \frac{13}{2} \)
…
\(\displaystyle \int_{2n}^{2n+1} x dx = \left[ \frac{x^2}{2} \right]_{2n}^{2n+1} = \frac{4n+1}{2} \)


\(\displaystyle \int_{\frac{1}{2}}^{\frac{5}{2}} x dx = 3 \)
\(\displaystyle \int_{\frac{9}{2}}^{\frac{13}{2}} x dx = 11 \)
\(\displaystyle \int_{\frac{17}{2}}^{\frac{21}{2}} x dx = 19 \)
\(\displaystyle \int_{\frac{25}{2}}^{\frac{29}{2}} x dx = 27 \)
…
\(\displaystyle \int_{\frac{4n+1}{2}}^{\frac{4n+5}{2}} x dx = \left[ \frac{x^2}{2} \right]_{\frac{4n+1}{2}}^{\frac{4n+5}{2}} = 4n+3 \)


\(\displaystyle \int_{3}^{11} x dx = 56 \)
\(\displaystyle \int_{19}^{27} x dx = 184 \)
\(\displaystyle \int_{35}^{43} x dx = 312 \)
\(\displaystyle \int_{51}^{59} x dx = 440 \)
…
\(\displaystyle \int_{4n+3}^{4n+11} x dx = \left[ \frac{x^2}{2} \right]_{4n+3}^{4n+11} = 32n+56 = 8\left( 4n+7 \right) \)