integral of t(t-1)^{12}

解答

\int t (t - 1)^{12} d t

\frac{t ^{14}}{14} - \frac{12 t ^{13}}{13} + \frac{11 t ^{12}}{2} - 20 t^{11} + \frac{99 t ^{10}}{2} - 88 t^{9} + \frac{231 t ^{8}}{2} - \frac{792 t ^{7}}{7} + \frac{165 t ^{6}}{2} - 44 t^{5} + \frac{33 t ^{4}}{2} - 4 t^{3} + \frac{t ^{2}}{2} + C

求解步骤

\int t (t - 1)^{12} d t

乘开

t (t - 1)^{12} : t^{13} - 12 t^{12} + 66 t^{11} - 220 t^{10} + 495 t^{9} - 792 t^{8} + 924 t^{7} - 792 t^{6} + 495 t^{5} - 220 t^{4} + 66 t^{3} - 12 t^{2} + t

= \int t^{13} - 12 t^{12} + 66 t^{11} - 220 t^{10} + 495 t^{9} - 792 t^{8} + 924 t^{7} - 792 t^{6} + 495 t^{5} - 220 t^{4} + 66 t^{3} - 12 t^{2} + t d t

使用积分加法定则:

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

= \int t^{13} d t - \int 12 t^{12} d t + \int 66 t^{11} d t - \int 220 t^{10} d t + \int 495 t^{9} d t - \int 792 t^{8} d t + \int 924 t^{7} d t - \int 792 t^{6} d t + \int 495 t^{5} d t - \int 220 t^{4} d t + \int 66 t^{3} d t - \int 12 t^{2} d t + \int t d t

\int t^{13} d t = \frac{t ^{14}}{14}

\int 12 t^{12} d t = \frac{12 t ^{13}}{13}

\int 66 t^{11} d t = \frac{11 t ^{12}}{2}

\int 220 t^{10} d t = 20 t^{11}

\int 495 t^{9} d t = \frac{99 t ^{10}}{2}

\int 792 t^{8} d t = 88 t^{9}

\int 924 t^{7} d t = \frac{231 t ^{8}}{2}

\int 792 t^{6} d t = \frac{792 t ^{7}}{7}

\int 495 t^{5} d t = \frac{165 t ^{6}}{2}

\int 220 t^{4} d t = 44 t^{5}

\int 66 t^{3} d t = \frac{33 t ^{4}}{2}

\int 12 t^{2} d t = 4 t^{3}

\int t d t = \frac{t ^{2}}{2}

= \frac{t ^{14}}{14} - \frac{12 t ^{13}}{13} + \frac{11 t ^{12}}{2} - 20 t^{11} + \frac{99 t ^{10}}{2} - 88 t^{9} + \frac{231 t ^{8}}{2} - \frac{792 t ^{7}}{7} + \frac{165 t ^{6}}{2} - 44 t^{5} + \frac{33 t ^{4}}{2} - 4 t^{3} + \frac{t ^{2}}{2}

解答补常数

= \frac{t ^{14}}{14} - \frac{12 t ^{13}}{13} + \frac{11 t ^{12}}{2} - 20 t^{11} + \frac{99 t ^{10}}{2} - 88 t^{9} + \frac{231 t ^{8}}{2} - \frac{792 t ^{7}}{7} + \frac{165 t ^{6}}{2} - 44 t^{5} + \frac{33 t ^{4}}{2} - 4 t^{3} + \frac{t ^{2}}{2} + C

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