What is the integral of x(5-x)^9 ?

The integral of x(5-x)^9 is -1/22 (-x+5)^{10}(2x+1)+C

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integral of x(5-x)^9

\int x (5 - x)^{9} d x

- \frac{1}{22} (- x + 5)^{10} (2 x + 1) + C

Solution steps

\int x (5 - x)^{9} 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}{9} \cdot \int - u^{\frac{2}{9}} + 5 u^{\frac{1}{9}} 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}{9} (- \int u^{\frac{2}{9}} d u + \int 5 u^{\frac{1}{9}} d u)

\int u^{\frac{2}{9}} d u = \frac{9}{11} u^{\frac{11}{9}}

\int 5 u^{\frac{1}{9}} d u = \frac{9}{2} u^{\frac{10}{9}}

= - \frac{1}{9} (- \frac{9}{11} u^{\frac{11}{9}} + \frac{9}{2} u^{\frac{10}{9}})

Substitute back

u = (5 - x)^{9}

= - \frac{1}{9} (- \frac{9}{11} ((5 - x)^{9})^{\frac{11}{9}} + \frac{9}{2} ((5 - x)^{9})^{\frac{10}{9}})

Simplify

- \frac{1}{9} (- \frac{9}{11} ((5 - x)^{9})^{\frac{11}{9}} + \frac{9}{2} ((5 - x)^{9})^{\frac{10}{9}}) : - \frac{1}{22} (- x + 5)^{10} (2 x + 1)

= - \frac{1}{22} (- x + 5)^{10} (2 x + 1)

Add a constant to the solution

= - \frac{1}{22} (- x + 5)^{10} (2 x + 1) + C

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What is the integral of x(5-x)^9 ?
The integral of x(5-x)^9 is -1/22 (-x+5)^{10}(2x+1)+C