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Popular >

integral of x^2(x-9)^{12}

Solution

\int x^{2} (x - 9)^{12} d x

Solution

\frac{x ^{15}}{15} - \frac{54 x ^{14}}{7} + \frac{5346 x ^{13}}{13} - 13365 x^{12} + 295245 x^{11} - \frac{23383404 x ^{10}}{5} + 54561276 x^{9} - 473513931 x^{8} + \frac{21308126895 x ^{7}}{7} - 14205417930 x^{6} + \frac{9 ^{10} \cdot 66 x ^{5}}{5} - 9^{11} \cdot 3 x^{4} + 3^{23} x^{3} + C

Solution steps

\int x^{2} (x - 9)^{12} d x

Expand

x^{2} (x - 9)^{12} : x^{14} - 108 x^{13} + 5346 x^{12} - 160380 x^{11} + 3247695 x^{10} - 46766808 x^{9} + 491051484 x^{8} - 3788111448 x^{7} + 21308126895 x^{6} - 85232507580 x^{5} + 9^{10} \cdot 66 x^{4} - 9^{11} \cdot 12 x^{3} + 9^{12} x^{2}

= \int x^{14} - 108 x^{13} + 5346 x^{12} - 160380 x^{11} + 3247695 x^{10} - 46766808 x^{9} + 491051484 x^{8} - 3788111448 x^{7} + 21308126895 x^{6} - 85232507580 x^{5} + 9^{10} \cdot 66 x^{4} - 9^{11} \cdot 12 x^{3} + 9^{12} x^{2} d x

Apply the Sum Rule:

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

= \int x^{14} d x - \int 108 x^{13} d x + \int 5346 x^{12} d x - \int 160380 x^{11} d x + \int 3247695 x^{10} d x - \int 46766808 x^{9} d x + \int 491051484 x^{8} d x - \int 3788111448 x^{7} d x + \int 21308126895 x^{6} d x - \int 85232507580 x^{5} d x + \int 9^{10} \cdot 66 x^{4} d x - \int 9^{11} \cdot 12 x^{3} d x + \int 9^{12} x^{2} d x

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

\int 108 x^{13} d x = \frac{54 x ^{14}}{7}

\int 5346 x^{12} d x = \frac{5346 x ^{13}}{13}

\int 160380 x^{11} d x = 13365 x^{12}

\int 3247695 x^{10} d x = 295245 x^{11}

\int 46766808 x^{9} d x = \frac{23383404 x ^{10}}{5}

\int 491051484 x^{8} d x = 54561276 x^{9}

\int 3788111448 x^{7} d x = 473513931 x^{8}

\int 21308126895 x^{6} d x = \frac{21308126895 x ^{7}}{7}

\int 85232507580 x^{5} d x = 14205417930 x^{6}

\int 9^{10} \cdot 66 x^{4} d x = \frac{9 ^{10} \cdot 66 x ^{5}}{5}

\int 9^{11} \cdot 12 x^{3} d x = 9^{11} \cdot 3 x^{4}

\int 9^{12} x^{2} d x = 3^{23} x^{3}

= \frac{x ^{15}}{15} - \frac{54 x ^{14}}{7} + \frac{5346 x ^{13}}{13} - 13365 x^{12} + 295245 x^{11} - \frac{23383404 x ^{10}}{5} + 54561276 x^{9} - 473513931 x^{8} + \frac{21308126895 x ^{7}}{7} - 14205417930 x^{6} + \frac{9 ^{10} \cdot 66 x ^{5}}{5} - 9^{11} \cdot 3 x^{4} + 3^{23} x^{3}

Add a constant to the solution

= \frac{x ^{15}}{15} - \frac{54 x ^{14}}{7} + \frac{5346 x ^{13}}{13} - 13365 x^{12} + 295245 x^{11} - \frac{23383404 x ^{10}}{5} + 54561276 x^{9} - 473513931 x^{8} + \frac{21308126895 x ^{7}}{7} - 14205417930 x^{6} + \frac{9 ^{10} \cdot 66 x ^{5}}{5} - 9^{11} \cdot 3 x^{4} + 3^{23} x^{3} + C

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Frequently Asked Questions (FAQ)

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