What is the integral of 5(1-x)^4 ?

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

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integral of 5(1-x)^4

\int 5 (1 - x)^{4} d x

- (1 - x)^{5} + C

Solution steps

\int 5 (1 - x)^{4} d x

Take the constant out:

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

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

Apply u-substitution

= 5 \cdot \int - u^{4} d u

Take the constant out:

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

= 5 (- \int u^{4} d u)

Apply the Power Rule

= 5 (- \frac{u ^{5}}{5})

Substitute back

u = 1 - x

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

Simplify

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

= - (1 - x)^{5}

Add a constant to the solution

= - (1 - x)^{5} + C