What is the integral from 0 to 1-x of 6x ?

The integral from 0 to 1-x of 6x is 3(-x+1)^2

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integral from 0 to 1-x of 6x

\int_{0}^{1 - x} 6 x d x

3 (- x + 1)^{2}

Solution steps

\int_{0}^{1 - x} 6 x d x

Take the constant out:

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

= 6 \cdot \int_{0}^{1 - x} x d x

Apply the Power Rule

: [\frac{x ^{2}}{2}]_{0}^{1 - x}

= 6 [\frac{x ^{2}}{2}]_{0}^{1 - x}

Compute the boundaries:

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

= 6 \cdot \frac{1}{2} (- x + 1)^{2}

Simplify

= 3 (- x + 1)^{2}

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What is the integral from 0 to 1-x of 6x ?
The integral from 0 to 1-x of 6x is 3(-x+1)^2