What is the integral of ((1-2x)/3)^3 ?

The integral of ((1-2x)/3)^3 is -1/216 (1-2x)^4+C

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integral of ((1-2x)/3)^3

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

- \frac{1}{216} (1 - 2 x)^{4} + C

Solution steps

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

Simplify

(\frac{1 - 2 x}{3})^{3} : \frac{1}{27} (1 - 2 x)^{3}

= \int \frac{1}{27} (1 - 2 x)^{3} d x

Take the constant out:

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

= \frac{1}{27} \cdot \int (1 - 2 x)^{3} d x

Apply u-substitution

: \int - \frac{u ^{3}}{2} d u

= \frac{1}{27} \cdot \int - \frac{u ^{3}}{2} d u

Take the constant out:

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

= \frac{1}{27} (- \frac{1}{2} \cdot \int u^{3} d u)

Apply the Power Rule

: \frac{u ^{4}}{4}

= \frac{1}{27} (- \frac{1}{2} \cdot \frac{u ^{4}}{4})

Substitute back

u = 1 - 2 x

= \frac{1}{27} (- \frac{1}{2} \cdot \frac{( 1 - 2 x ) ^{4}}{4})

Simplify

\frac{1}{27} (- \frac{1}{2} \cdot \frac{( 1 - 2 x ) ^{4}}{4}) : - \frac{1}{216} (1 - 2 x)^{4}

= - \frac{1}{216} (1 - 2 x)^{4}

Add a constant to the solution

= - \frac{1}{216} (1 - 2 x)^{4} + C

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What is the integral of ((1-2x)/3)^3 ?
The integral of ((1-2x)/3)^3 is -1/216 (1-2x)^4+C