What is the integral of x*(x+5)^8 ?

The integral of x*(x+5)^8 is ((x+5)^{10}}{10}-\frac{5(x+5)^9)/9+C

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integral of x*(x+5)^8

integral

x \cdot (x + 5)^{8}

\frac{( x + 5 ) ^{10}}{10} - \frac{5 ( x + 5 ) ^{9}}{9} + C

Solution steps

\int x (x + 5)^{8} d x

Apply u-substitution

= \int (u - 5) u^{8} d u

Expand

(u - 5) u^{8} : u^{9} - 5 u^{8}

= \int u^{9} - 5 u^{8} 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

= \int u^{9} d u - \int 5 u^{8} d u

\int u^{9} d u = \frac{u ^{10}}{10}

\int 5 u^{8} d u = \frac{5 u ^{9}}{9}

= \frac{u ^{10}}{10} - \frac{5 u ^{9}}{9}

Substitute back

u = x + 5

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

Add a constant to the solution

= \frac{( x + 5 ) ^{10}}{10} - \frac{5 ( x + 5 ) ^{9}}{9} + C

What is the integral of x*(x+5)^8 ?
The integral of x*(x+5)^8 is ((x+5)^{10}}{10}-\frac{5(x+5)^9)/9+C