What is the integral from-1 to 2 of-(1-x)^2 ?

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

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integral from-1 to 2 of-(1-x)^2

\int_{- 1}^{2} - (1 - x)^{2} d x

- 3

Solution steps

\int_{- 1}^{2} - (1 - x)^{2} d x

Take the constant out:

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

= - \int_{- 1}^{2} (1 - x)^{2} d x

Expand

(1 - x)^{2} : 1 - 2 x + x^{2}

= - \int_{- 1}^{2} 1 - 2 x + x^{2} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - (\int_{- 1}^{2} 1 d x - \int_{- 1}^{2} 2 x d x + \int_{- 1}^{2} x^{2} d x)

\int_{- 1}^{2} 1 d x = 3

\int_{- 1}^{2} 2 x d x = 3

\int_{- 1}^{2} x^{2} d x = 3

= - (3 - 3 + 3)

Simplify

= - 3

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