What is the integral of 24(1-x)^5 ?

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

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integral of 24(1-x)^5

\int 24 (1 - x)^{5} d x

- 4 (- x + 1)^{6} + C

Solution steps

\int 24 (1 - x)^{5} d x

Take the constant out:

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

= 24 \cdot \int (1 - x)^{5} d x

Apply u-substitution

: \int - u^{5} d u

= 24 \cdot \int - u^{5} d u

Take the constant out:

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

= 24 (- \int u^{5} d u)

Apply the Power Rule

: \frac{u ^{6}}{6}

= 24 (- \frac{u ^{6}}{6})

Substitute back

u = 1 - x

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

Simplify

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

= - 4 (- x + 1)^{6}

Add a constant to the solution

= - 4 (- x + 1)^{6} + C

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