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

The integral of x(5-x)^7 is -1/72 (-x+5)^8(8x+5)+C

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integral of x(5-x)^7

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

- \frac{1}{72} (- x + 5)^{8} (8 x + 5) + C

Solution steps

\int x (5 - x)^{7} 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}{7} \cdot \int - u^{\frac{2}{7}} + 5 u^{\frac{1}{7}} 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}{7} (- \int u^{\frac{2}{7}} d u + \int 5 u^{\frac{1}{7}} d u)

\int u^{\frac{2}{7}} d u = \frac{7}{9} u^{\frac{9}{7}}

\int 5 u^{\frac{1}{7}} d u = \frac{35}{8} u^{\frac{8}{7}}

= - \frac{1}{7} (- \frac{7}{9} u^{\frac{9}{7}} + \frac{35}{8} u^{\frac{8}{7}})

Substitute back

u = (5 - x)^{7}

= - \frac{1}{7} (- \frac{7}{9} ((5 - x)^{7})^{\frac{9}{7}} + \frac{35}{8} ((5 - x)^{7})^{\frac{8}{7}})

Simplify

- \frac{1}{7} (- \frac{7}{9} ((5 - x)^{7})^{\frac{9}{7}} + \frac{35}{8} ((5 - x)^{7})^{\frac{8}{7}}) : - \frac{1}{72} (- x + 5)^{8} (8 x + 5)

= - \frac{1}{72} (- x + 5)^{8} (8 x + 5)

Add a constant to the solution

= - \frac{1}{72} (- x + 5)^{8} (8 x + 5) + C

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What is the integral of x(5-x)^7 ?
The integral of x(5-x)^7 is -1/72 (-x+5)^8(8x+5)+C