What is the integral of 72x^7(1-x) ?

The integral of 72x^7(1-x) is 9x^8-8x^9+C

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integral of 72x^7(1-x)

\int 72 x^{7} (1 - x) d x

9 x^{8} - 8 x^{9} + C

Solution steps

\int 72 x^{7} (1 - x) d x

Take the constant out:

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

= 72 \cdot \int x^{7} (1 - x) d x

Apply Integration By Parts

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

= 72 (\frac{1}{8} x^{8} (1 - x) - \int - \frac{x ^{8}}{8} d x)

\int - \frac{x ^{8}}{8} d x = - \frac{x ^{9}}{72}

= 72 (\frac{1}{8} x^{8} (1 - x) - (- \frac{x ^{9}}{72}))

Simplify

72 (\frac{1}{8} x^{8} (1 - x) - (- \frac{x ^{9}}{72})) : 9 x^{8} - 8 x^{9}

= 9 x^{8} - 8 x^{9}

Add a constant to the solution

= 9 x^{8} - 8 x^{9} + C

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