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

The integral of (5-4x)^5 is -1/24 (5-4x)^6+C

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integral of (5-4x)^5

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

- \frac{1}{24} (5 - 4 x)^{6} + C

Solution steps

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

Apply u-substitution

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

Apply the Power Rule

= - \frac{1}{4} \cdot \frac{u ^{6}}{6}

Substitute back

u = 5 - 4 x

= - \frac{1}{4} \cdot \frac{( 5 - 4 x ) ^{6}}{6}

Simplify

- \frac{1}{4} \cdot \frac{( 5 - 4 x ) ^{6}}{6} : - \frac{1}{24} (5 - 4 x)^{6}

= - \frac{1}{24} (5 - 4 x)^{6}

Add a constant to the solution

= - \frac{1}{24} (5 - 4 x)^{6} + C