What is the integral of x(7-x)^5 ?

The integral of x(7-x)^5 is (16807x^2)/2-(12005x^3)/3+(1715x^4)/2-98x^5+(35x^6)/6-(x^7)/7+C

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integral of x(7-x)^5

\int x (7 - x)^{5} d x

\frac{16807 x ^{2}}{2} - \frac{12005 x ^{3}}{3} + \frac{1715 x ^{4}}{2} - 98 x^{5} + \frac{35 x ^{6}}{6} - \frac{x ^{7}}{7} + C

Solution steps

\int x (7 - x)^{5} d x

Expand

x (7 - x)^{5} : 16807 x - 12005 x^{2} + 3430 x^{3} - 490 x^{4} + 35 x^{5} - x^{6}

= \int 16807 x - 12005 x^{2} + 3430 x^{3} - 490 x^{4} + 35 x^{5} - x^{6} 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 16807 x d x - \int 12005 x^{2} d x + \int 3430 x^{3} d x - \int 490 x^{4} d x + \int 35 x^{5} d x - \int x^{6} d x

\int 16807 x d x = \frac{16807 x ^{2}}{2}

\int 12005 x^{2} d x = \frac{12005 x ^{3}}{3}

\int 3430 x^{3} d x = \frac{1715 x ^{4}}{2}

\int 490 x^{4} d x = 98 x^{5}

\int 35 x^{5} d x = \frac{35 x ^{6}}{6}

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

= \frac{16807 x ^{2}}{2} - \frac{12005 x ^{3}}{3} + \frac{1715 x ^{4}}{2} - 98 x^{5} + \frac{35 x ^{6}}{6} - \frac{x ^{7}}{7}

Add a constant to the solution

= \frac{16807 x ^{2}}{2} - \frac{12005 x ^{3}}{3} + \frac{1715 x ^{4}}{2} - 98 x^{5} + \frac{35 x ^{6}}{6} - \frac{x ^{7}}{7} + C

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What is the integral of x(7-x)^5 ?
The integral of x(7-x)^5 is (16807x^2)/2-(12005x^3)/3+(1715x^4)/2-98x^5+(35x^6)/6-(x^7)/7+C