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

The integral from-2 to 2 of xsqrt(2-x) is -32/15

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integral from-2 to 2 of xsqrt(2-x)

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

- \frac{32}{15}

Decimal

- 2.13333 \dots

Solution steps

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

Apply u-substitution

= \int_{4}^{0} - (- u + 2) u d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{0}^{4} - (- u + 2) u d u

Take the constant out:

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

= - (- \int_{0}^{4} (- u + 2) u d u)

Expand

(- u + 2) u : - u^{\frac{3}{2}} + 2 u

= - (- \int_{0}^{4} - u^{\frac{3}{2}} + 2 u d u)

Apply the Sum Rule:

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

= - (- (- \int_{0}^{4} u^{\frac{3}{2}} d u + \int_{0}^{4} 2 u d u))

\int_{0}^{4} u^{\frac{3}{2}} d u = \frac{64}{5}

\int_{0}^{4} 2 u d u = \frac{32}{3}

= - (- (- \frac{64}{5} + \frac{32}{3}))

Simplify

- (- (- \frac{64}{5} + \frac{32}{3})) : - \frac{32}{15}

= - \frac{32}{15}

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What is the integral from-2 to 2 of xsqrt(2-x) ?
The integral from-2 to 2 of xsqrt(2-x) is -32/15