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

The integral of x(x-6)^5 is (x^7)/7-5x^6+72x^5-540x^4+2160x^3-3888x^2+C

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integral of x(x-6)^5

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

\frac{x ^{7}}{7} - 5 x^{6} + 72 x^{5} - 540 x^{4} + 2160 x^{3} - 3888 x^{2} + C

Solution steps

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

Expand

x (x - 6)^{5} : x^{6} - 30 x^{5} + 360 x^{4} - 2160 x^{3} + 6480 x^{2} - 7776 x

= \int x^{6} - 30 x^{5} + 360 x^{4} - 2160 x^{3} + 6480 x^{2} - 7776 x 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^{6} d x - \int 30 x^{5} d x + \int 360 x^{4} d x - \int 2160 x^{3} d x + \int 6480 x^{2} d x - \int 7776 x d x

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

\int 30 x^{5} d x = 5 x^{6}

\int 360 x^{4} d x = 72 x^{5}

\int 2160 x^{3} d x = 540 x^{4}

\int 6480 x^{2} d x = 2160 x^{3}

\int 7776 x d x = 3888 x^{2}

= \frac{x ^{7}}{7} - 5 x^{6} + 72 x^{5} - 540 x^{4} + 2160 x^{3} - 3888 x^{2}

Add a constant to the solution

= \frac{x ^{7}}{7} - 5 x^{6} + 72 x^{5} - 540 x^{4} + 2160 x^{3} - 3888 x^{2} + C

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What is the integral of x(x-6)^5 ?
The integral of x(x-6)^5 is (x^7)/7-5x^6+72x^5-540x^4+2160x^3-3888x^2+C