What is the integral of 90x^8(1-x) ?

The integral of 90x^8(1-x) is 10x^9-9x^{10}+C

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integral of 90x^8(1-x)

\int 90 x^{8} (1 - x) d x

10 x^{9} - 9 x^{10} + C

Solution steps

\int 90 x^{8} (1 - x) d x

Take the constant out:

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

= 90 \cdot \int x^{8} (1 - x) d x

Apply Integration By Parts

: \frac{1}{9} x^{9} (1 - x) - \int - \frac{x ^{9}}{9} d x

= 90 (\frac{1}{9} x^{9} (1 - x) - \int - \frac{x ^{9}}{9} d x)

\int - \frac{x ^{9}}{9} d x = - \frac{x ^{10}}{90}

= 90 (\frac{1}{9} x^{9} (1 - x) - (- \frac{x ^{10}}{90}))

Simplify

90 (\frac{1}{9} x^{9} (1 - x) - (- \frac{x ^{10}}{90})) : 10 x^{9} - 9 x^{10}

= 10 x^{9} - 9 x^{10}

Add a constant to the solution

= 10 x^{9} - 9 x^{10} + C

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What is the integral of 90x^8(1-x) ?
The integral of 90x^8(1-x) is 10x^9-9x^{10}+C