What is the integral of ((2-x)^3)/3 ?

The integral of ((2-x)^3)/3 is -1/12 (2-x)^4+C

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integral of ((2-x)^3)/3

\int \frac{( 2 - x ) ^{3}}{3} d x

- \frac{1}{12} (2 - x)^{4} + C

Solution steps

\int \frac{( 2 - x ) ^{3}}{3} d x

Take the constant out:

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

= \frac{1}{3} \cdot \int (2 - x)^{3} d x

Apply u-substitution

= \frac{1}{3} \cdot \int - u^{3} d u

Take the constant out:

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

= \frac{1}{3} (- \int u^{3} d u)

Apply the Power Rule

= \frac{1}{3} (- \frac{u ^{4}}{4})

Substitute back

u = 2 - x

= \frac{1}{3} (- \frac{( 2 - x ) ^{4}}{4})

Simplify

\frac{1}{3} (- \frac{( 2 - x ) ^{4}}{4}) : - \frac{1}{12} (2 - x)^{4}

= - \frac{1}{12} (2 - x)^{4}

Add a constant to the solution

= - \frac{1}{12} (2 - x)^{4} + C

What is the integral of ((2-x)^3)/3 ?
The integral of ((2-x)^3)/3 is -1/12 (2-x)^4+C