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

The integral from-1 to 1 of (x-1)sqrt(1-x^2) is -pi/2

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

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

- \frac{π}{2}

Decimal

- 1.57079 \dots

Solution steps

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

Expand

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

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

\int_{- 1}^{1} x 1 - x^{2} d x = 0

\int_{- 1}^{1} 1 - x^{2} d x = \frac{π}{2}

= 0 - \frac{π}{2}

Simplify

= - \frac{π}{2}

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\int_{- 8}^{9} (1 - x) d x

\int_{0}^{π} sin^{2} (x) sin (x) d x

\int_{- π}^{π} sin (m) x sin (n) x d x

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