What is the integral of x(7-x)^{11} ?

The integral of x(7-x)^{11} is -1/156 (-x+7)^{12}(12x+7)+C

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integral of x(7-x)^{11}

\int x (7 - x)^{11} d x

- \frac{1}{156} (- x + 7)^{12} (12 x + 7) + C

Solution steps

\int x (7 - x)^{11} d x

Apply u-substitution

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

Expand

= - \frac{1}{11} \cdot \int - u^{\frac{2}{11}} + 7 u^{\frac{1}{11}} d u

Apply the Sum Rule:

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

= - \frac{1}{11} (- \int u^{\frac{2}{11}} d u + \int 7 u^{\frac{1}{11}} d u)

\int u^{\frac{2}{11}} d u = \frac{11}{13} u^{\frac{13}{11}}

\int 7 u^{\frac{1}{11}} d u = \frac{77}{12} u^{\frac{12}{11}}

= - \frac{1}{11} (- \frac{11}{13} u^{\frac{13}{11}} + \frac{77}{12} u^{\frac{12}{11}})

Substitute back

u = (7 - x)^{11}

= - \frac{1}{11} (- \frac{11}{13} ((7 - x)^{11})^{\frac{13}{11}} + \frac{77}{12} ((7 - x)^{11})^{\frac{12}{11}})

Simplify

- \frac{1}{11} (- \frac{11}{13} ((7 - x)^{11})^{\frac{13}{11}} + \frac{77}{12} ((7 - x)^{11})^{\frac{12}{11}}) : - \frac{1}{156} (- x + 7)^{12} (12 x + 7)

= - \frac{1}{156} (- x + 7)^{12} (12 x + 7)

Add a constant to the solution

= - \frac{1}{156} (- x + 7)^{12} (12 x + 7) + C

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What is the integral of x(7-x)^{11} ?
The integral of x(7-x)^{11} is -1/156 (-x+7)^{12}(12x+7)+C