What is the integral of (x^2-6x)^5(x-3) ?

The integral of (x^2-6x)^5(x-3) is (x^{12})/(12)-3x^{11}+45x^{10}-360x^9+1620x^8-3888x^7+3888x^6+C

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integral of (x^2-6x)^5(x-3)

\int (x^{2} - 6 x)^{5} (x - 3) d x

\frac{x ^{12}}{12} - 3 x^{11} + 45 x^{10} - 360 x^{9} + 1620 x^{8} - 3888 x^{7} + 3888 x^{6} + C

Solution steps

\int (x^{2} - 6 x)^{5} (x - 3) d x

Expand

(x^{2} - 6 x)^{5} (x - 3) : x^{11} - 33 x^{10} + 450 x^{9} - 3240 x^{8} + 12960 x^{7} - 27216 x^{6} + 23328 x^{5}

= \int x^{11} - 33 x^{10} + 450 x^{9} - 3240 x^{8} + 12960 x^{7} - 27216 x^{6} + 23328 x^{5} 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^{11} d x - \int 33 x^{10} d x + \int 450 x^{9} d x - \int 3240 x^{8} d x + \int 12960 x^{7} d x - \int 27216 x^{6} d x + \int 23328 x^{5} d x

\int x^{11} d x = \frac{x ^{12}}{12}

\int 33 x^{10} d x = 3 x^{11}

\int 450 x^{9} d x = 45 x^{10}

\int 3240 x^{8} d x = 360 x^{9}

\int 12960 x^{7} d x = 1620 x^{8}

\int 27216 x^{6} d x = 3888 x^{7}

\int 23328 x^{5} d x = 3888 x^{6}

= \frac{x ^{12}}{12} - 3 x^{11} + 45 x^{10} - 360 x^{9} + 1620 x^{8} - 3888 x^{7} + 3888 x^{6}

Add a constant to the solution

= \frac{x ^{12}}{12} - 3 x^{11} + 45 x^{10} - 360 x^{9} + 1620 x^{8} - 3888 x^{7} + 3888 x^{6} + C

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What is the integral of (x^2-6x)^5(x-3) ?
The integral of (x^2-6x)^5(x-3) is (x^{12})/(12)-3x^{11}+45x^{10}-360x^9+1620x^8-3888x^7+3888x^6+C