What is the integral of (5-4x)^7 ?

The integral of (5-4x)^7 is -1/32 (5-4x)^8+C

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integral of (5-4x)^7

\int (5 - 4 x)^{7} d x

- \frac{1}{32} (5 - 4 x)^{8} + C

Solution steps

\int (5 - 4 x)^{7} d x

Apply u-substitution

: \int - \frac{u ^{7}}{4} d u

= \int - \frac{u ^{7}}{4} d u

Take the constant out:

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

= - \frac{1}{4} \cdot \int u^{7} d u

Apply the Power Rule

: \frac{u ^{8}}{8}

= - \frac{1}{4} \cdot \frac{u ^{8}}{8}

Substitute back

u = 5 - 4 x

= - \frac{1}{4} \cdot \frac{( 5 - 4 x ) ^{8}}{8}

Simplify

- \frac{1}{4} \cdot \frac{( 5 - 4 x ) ^{8}}{8} : - \frac{1}{32} (5 - 4 x)^{8}

= - \frac{1}{32} (5 - 4 x)^{8}

Add a constant to the solution

= - \frac{1}{32} (5 - 4 x)^{8} + C

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