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

The integral of (5-4x)^4 is -1/20 (5-4x)^5+C

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integral of (5-4x)^4

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

- \frac{1}{20} (5 - 4 x)^{5} + C

Solution steps

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

Apply u-substitution

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

Apply the Power Rule

= - \frac{1}{4} \cdot \frac{u ^{5}}{5}

Substitute back

u = 5 - 4 x

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

Simplify

- \frac{1}{4} \cdot \frac{( 5 - 4 x ) ^{5}}{5} : - \frac{1}{20} (5 - 4 x)^{5}

= - \frac{1}{20} (5 - 4 x)^{5}

Add a constant to the solution

= - \frac{1}{20} (5 - 4 x)^{5} + C