What is the integral from 0 to 1 of xsqrt(2-x) ?

The integral from 0 to 1 of xsqrt(2-x) is (16sqrt(2)-14)/(15)

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integral from 0 to 1 of xsqrt(2-x)

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

\frac{16 2 - 14}{15}

Decimal

0.57516 \dots

Solution steps

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

Apply u-substitution

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

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

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

Take the constant out:

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

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

Expand

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

= - (- \int_{1}^{2} - 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_{1}^{2} u^{\frac{3}{2}} d u + \int_{1}^{2} 2 u d u))

\int_{1}^{2} u^{\frac{3}{2}} d u = \frac{8 2 - 2}{5}

\int_{1}^{2} 2 u d u = \frac{2 ( 4 2 - 2 )}{3}

= - (- (- \frac{8 2 - 2}{5} + \frac{2 ( 4 2 - 2 )}{3}))

Simplify

- (- (- \frac{8 2 - 2}{5} + \frac{2 ( 4 2 - 2 )}{3})) : \frac{16 2 - 14}{15}

= \frac{16 2 - 14}{15}

What is the integral from 0 to 1 of xsqrt(2-x) ?
The integral from 0 to 1 of xsqrt(2-x) is (16sqrt(2)-14)/(15)