integral from 6 to 5 of x(x-5)^{12}

解

\int_{6}^{5} x (x - 5)^{12} d x

- (\frac{6 ^{14}}{14} - \frac{60 \cdot 6 ^{13}}{13} + \frac{275 \cdot 6 ^{12}}{2} - 2500 \cdot 6^{11} + \frac{61875 \cdot 6 ^{10}}{2} - 275000 \cdot 6^{9} + \frac{3609375 \cdot 6 ^{8}}{2} - \frac{61875000 \cdot 6 ^{7}}{7} + \frac{64453125 \cdot 6 ^{6}}{2} - 85937500 \cdot 6^{5} + \frac{322265625 \cdot 6 ^{4}}{2} - 195312500 \cdot 6^{3} + \frac{244140625 \cdot 6 ^{2}}{2} - (\frac{5 ^{14}}{14} - \frac{60 \cdot 5 ^{13}}{13} + \frac{275 \cdot 5 ^{12}}{2} - 2500 \cdot 5^{11} + \frac{61875 \cdot 5 ^{10}}{2} - 275000 \cdot 5^{9} + \frac{3609375 \cdot 5 ^{8}}{2} - \frac{61875000 \cdot 5 ^{7}}{7} + \frac{64453125 \cdot 5 ^{6}}{2} - 85937500 \cdot 5^{5} + \frac{322265625 \cdot 5 ^{4}}{2} - 195312500 \cdot 5^{3} + \frac{244140625 \cdot 5 ^{2}}{2}))

十進法表記

- 0.45609 \dots

解答ステップ

\int_{6}^{5} x (x - 5)^{12} d x

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{5}^{6} x (x - 5)^{12} d x

\int_{5}^{6} x (x - 5)^{12} d x = \frac{6 ^{14}}{14} - \frac{60 \cdot 6 ^{13}}{13} + \frac{275 \cdot 6 ^{12}}{2} - 2500 \cdot 6^{11} + \frac{61875 \cdot 6 ^{10}}{2} - 275000 \cdot 6^{9} + \frac{3609375 \cdot 6 ^{8}}{2} - \frac{61875000 \cdot 6 ^{7}}{7} + \frac{64453125 \cdot 6 ^{6}}{2} - 85937500 \cdot 6^{5} + \frac{322265625 \cdot 6 ^{4}}{2} - 195312500 \cdot 6^{3} + \frac{244140625 \cdot 6 ^{2}}{2} - (\frac{5 ^{14}}{14} - \frac{60 \cdot 5 ^{13}}{13} + \frac{275 \cdot 5 ^{12}}{2} - 2500 \cdot 5^{11} + \frac{61875 \cdot 5 ^{10}}{2} - 275000 \cdot 5^{9} + \frac{3609375 \cdot 5 ^{8}}{2} - \frac{61875000 \cdot 5 ^{7}}{7} + \frac{64453125 \cdot 5 ^{6}}{2} - 85937500 \cdot 5^{5} + \frac{322265625 \cdot 5 ^{4}}{2} - 195312500 \cdot 5^{3} + \frac{244140625 \cdot 5 ^{2}}{2})

= - (\frac{6 ^{14}}{14} - \frac{60 \cdot 6 ^{13}}{13} + \frac{275 \cdot 6 ^{12}}{2} - 2500 \cdot 6^{11} + \frac{61875 \cdot 6 ^{10}}{2} - 275000 \cdot 6^{9} + \frac{3609375 \cdot 6 ^{8}}{2} - \frac{61875000 \cdot 6 ^{7}}{7} + \frac{64453125 \cdot 6 ^{6}}{2} - 85937500 \cdot 6^{5} + \frac{322265625 \cdot 6 ^{4}}{2} - 195312500 \cdot 6^{3} + \frac{244140625 \cdot 6 ^{2}}{2} - (\frac{5 ^{14}}{14} - \frac{60 \cdot 5 ^{13}}{13} + \frac{275 \cdot 5 ^{12}}{2} - 2500 \cdot 5^{11} + \frac{61875 \cdot 5 ^{10}}{2} - 275000 \cdot 5^{9} + \frac{3609375 \cdot 5 ^{8}}{2} - \frac{61875000 \cdot 5 ^{7}}{7} + \frac{64453125 \cdot 5 ^{6}}{2} - 85937500 \cdot 5^{5} + \frac{322265625 \cdot 5 ^{4}}{2} - 195312500 \cdot 5^{3} + \frac{244140625 \cdot 5 ^{2}}{2}))