integral of x^2(x-8)^{13}

해법

\int x^{2} (x - 8)^{13} d x

\frac{x ^{16}}{16} - \frac{104 x ^{15}}{15} + \frac{2496 x ^{14}}{7} - 11264 x^{13} + \frac{732160 x ^{12}}{3} - 3833856 x^{11} + \frac{224919552 x ^{10}}{5} - \frac{1199570944 x ^{9}}{3} + 2699034624 x^{8} - \frac{95965675520 x ^{7}}{7} + \frac{8 ^{10} \cdot 143 x ^{6}}{3} - \frac{8 ^{11} \cdot 78 x ^{5}}{5} + 2^{34} \cdot 13 x^{4} - \frac{8 ^{13} x ^{3}}{3} + C

솔루션 단계

\int x^{2} (x - 8)^{13} d x

x^{2} (x - 8)^{13}

확대한다:

x^{15} - 104 x^{14} + 4992 x^{13} - 146432 x^{12} + 2928640 x^{11} - 42172416 x^{10} + 449839104 x^{9} - 3598712832 x^{8} + 21592276992 x^{7} - 95965675520 x^{6} + 8^{10} \cdot 286 x^{5} - 8^{11} \cdot 78 x^{4} + 8^{12} \cdot 13 x^{3} - 8^{13} x^{2}

= \int x^{15} - 104 x^{14} + 4992 x^{13} - 146432 x^{12} + 2928640 x^{11} - 42172416 x^{10} + 449839104 x^{9} - 3598712832 x^{8} + 21592276992 x^{7} - 95965675520 x^{6} + 8^{10} \cdot 286 x^{5} - 8^{11} \cdot 78 x^{4} + 8^{12} \cdot 13 x^{3} - 8^{13} x^{2} d x

합계 규칙 적용:

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

= \int x^{15} d x - \int 104 x^{14} d x + \int 4992 x^{13} d x - \int 146432 x^{12} d x + \int 2928640 x^{11} d x - \int 42172416 x^{10} d x + \int 449839104 x^{9} d x - \int 3598712832 x^{8} d x + \int 21592276992 x^{7} d x - \int 95965675520 x^{6} d x + \int 8^{10} \cdot 286 x^{5} d x - \int 8^{11} \cdot 78 x^{4} d x + \int 8^{12} \cdot 13 x^{3} d x - \int 8^{13} x^{2} d x

\int x^{15} d x = \frac{x ^{16}}{16}

\int 104 x^{14} d x = \frac{104 x ^{15}}{15}

\int 4992 x^{13} d x = \frac{2496 x ^{14}}{7}

\int 146432 x^{12} d x = 11264 x^{13}

\int 2928640 x^{11} d x = \frac{732160 x ^{12}}{3}

\int 42172416 x^{10} d x = 3833856 x^{11}

\int 449839104 x^{9} d x = \frac{224919552 x ^{10}}{5}

\int 3598712832 x^{8} d x = \frac{1199570944 x ^{9}}{3}

\int 21592276992 x^{7} d x = 2699034624 x^{8}

\int 95965675520 x^{6} d x = \frac{95965675520 x ^{7}}{7}

\int 8^{10} \cdot 286 x^{5} d x = \frac{8 ^{10} \cdot 143 x ^{6}}{3}

\int 8^{11} \cdot 78 x^{4} d x = \frac{8 ^{11} \cdot 78 x ^{5}}{5}

\int 8^{12} \cdot 13 x^{3} d x = 2^{34} \cdot 13 x^{4}

\int 8^{13} x^{2} d x = \frac{8 ^{13} x ^{3}}{3}

= \frac{x ^{16}}{16} - \frac{104 x ^{15}}{15} + \frac{2496 x ^{14}}{7} - 11264 x^{13} + \frac{732160 x ^{12}}{3} - 3833856 x^{11} + \frac{224919552 x ^{10}}{5} - \frac{1199570944 x ^{9}}{3} + 2699034624 x^{8} - \frac{95965675520 x ^{7}}{7} + \frac{8 ^{10} \cdot 143 x ^{6}}{3} - \frac{8 ^{11} \cdot 78 x ^{5}}{5} + 2^{34} \cdot 13 x^{4} - \frac{8 ^{13} x ^{3}}{3}

솔루션에 상수 추가

= \frac{x ^{16}}{16} - \frac{104 x ^{15}}{15} + \frac{2496 x ^{14}}{7} - 11264 x^{13} + \frac{732160 x ^{12}}{3} - 3833856 x^{11} + \frac{224919552 x ^{10}}{5} - \frac{1199570944 x ^{9}}{3} + 2699034624 x^{8} - \frac{95965675520 x ^{7}}{7} + \frac{8 ^{10} \cdot 143 x ^{6}}{3} - \frac{8 ^{11} \cdot 78 x ^{5}}{5} + 2^{34} \cdot 13 x^{4} - \frac{8 ^{13} x ^{3}}{3} + C