What is the integral of (7-5x)^8 ?

The integral of (7-5x)^8 is -1/45 (7-5x)^9+C

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integral of (7-5x)^8

\int (7 - 5 x)^{8} d x

- \frac{1}{45} (7 - 5 x)^{9} + C

Solution steps

\int (7 - 5 x)^{8} d x

Apply u-substitution

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

Take the constant out:

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

= - \frac{1}{5} \cdot \int u^{8} d u

Apply the Power Rule

= - \frac{1}{5} \cdot \frac{u ^{9}}{9}

Substitute back

u = 7 - 5 x

= - \frac{1}{5} \cdot \frac{( 7 - 5 x ) ^{9}}{9}

Simplify

- \frac{1}{5} \cdot \frac{( 7 - 5 x ) ^{9}}{9} : - \frac{1}{45} (7 - 5 x)^{9}

= - \frac{1}{45} (7 - 5 x)^{9}

Add a constant to the solution

= - \frac{1}{45} (7 - 5 x)^{9} + C