integral from 5 to s of (t-t^2)^8

解

\int_{5}^{s} (t - t^{2})^{8} d t

\frac{s ^{9} - 1953125}{9} - 8 (\frac{s ^{10}}{10} - \frac{1953125}{2}) + \frac{28 ( s ^{11} - 48828125 )}{11} - \frac{14 ( s ^{12} - 244140625 )}{3} + \frac{70 ( s ^{13} - 5 ^{13} )}{13} - 4 (s^{14} - 5^{14}) + 28 (\frac{s ^{15}}{15} - \frac{5 ^{14}}{3}) - \frac{s ^{16} - 5 ^{16}}{2} + \frac{s ^{17} - 5 ^{17}}{17}

解答ステップ

\int_{5}^{s} (t - t^{2})^{8} d t

拡張

(t - t^{2})^{8} : t^{8} - 8 t^{9} + 28 t^{10} - 56 t^{11} + 70 t^{12} - 56 t^{13} + 28 t^{14} - 8 t^{15} + t^{16}

= \int_{5}^{s} t^{8} - 8 t^{9} + 28 t^{10} - 56 t^{11} + 70 t^{12} - 56 t^{13} + 28 t^{14} - 8 t^{15} + t^{16} d t

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{5}^{s} t^{8} d t - \int_{5}^{s} 8 t^{9} d t + \int_{5}^{s} 28 t^{10} d t - \int_{5}^{s} 56 t^{11} d t + \int_{5}^{s} 70 t^{12} d t - \int_{5}^{s} 56 t^{13} d t + \int_{5}^{s} 28 t^{14} d t - \int_{5}^{s} 8 t^{15} d t + \int_{5}^{s} t^{16} d t

\int_{5}^{s} t^{8} d t = \frac{s ^{9} - 1953125}{9}

\int_{5}^{s} 8 t^{9} d t = 8 (\frac{s ^{10}}{10} - \frac{1953125}{2})

\int_{5}^{s} 28 t^{10} d t = \frac{28 ( s ^{11} - 48828125 )}{11}

\int_{5}^{s} 56 t^{11} d t = \frac{14 ( s ^{12} - 244140625 )}{3}

\int_{5}^{s} 70 t^{12} d t = \frac{70 ( s ^{13} - 5 ^{13} )}{13}

\int_{5}^{s} 56 t^{13} d t = 4 (s^{14} - 5^{14})

\int_{5}^{s} 28 t^{14} d t = 28 (\frac{s ^{15}}{15} - \frac{5 ^{14}}{3})

\int_{5}^{s} 8 t^{15} d t = \frac{s ^{16} - 5 ^{16}}{2}

\int_{5}^{s} t^{16} d t = \frac{s ^{17} - 5 ^{17}}{17}

= \frac{s ^{9} - 1953125}{9} - 8 (\frac{s ^{10}}{10} - \frac{1953125}{2}) + \frac{28 ( s ^{11} - 48828125 )}{11} - \frac{14 ( s ^{12} - 244140625 )}{3} + \frac{70 ( s ^{13} - 5 ^{13} )}{13} - 4 (s^{14} - 5^{14}) + 28 (\frac{s ^{15}}{15} - \frac{5 ^{14}}{3}) - \frac{s ^{16} - 5 ^{16}}{2} + \frac{s ^{17} - 5 ^{17}}{17}