What is the integral from 0 to 5 of xsqrt(5-x) ?

The integral from 0 to 5 of xsqrt(5-x) is -10sqrt(5)+(50sqrt(5))/3

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integral from 0 to 5 of xsqrt(5-x)

\int_{0}^{5} x 5 - x d x

- 105 + \frac{50 5}{3}

Decimal

14.90711 \dots

Solution steps

\int_{0}^{5} x 5 - x d x

Apply u-substitution

: \int_{5}^{0} - (- u + 5) u d u

= \int_{5}^{0} - (- u + 5) u d u

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

= - \int_{0}^{5} - (- u + 5) u d u

Take the constant out:

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

= - (- \int_{0}^{5} (- u + 5) u d u)

Expand

(- u + 5) u : - u^{\frac{3}{2}} + 5 u

= - (- \int_{0}^{5} - u^{\frac{3}{2}} + 5 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}^{5} u^{\frac{3}{2}} d u + \int_{0}^{5} 5 u d u))

\int_{0}^{5} u^{\frac{3}{2}} d u = 105

\int_{0}^{5} 5 u d u = \frac{50 5}{3}

= - (- (- 105 + \frac{50 5}{3}))

Simplify

= - 105 + \frac{50 5}{3}

x^{2} y^{^{''}} + 2 x y^{^{'}} - 6 y = 0, y (1) = 0

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What is the integral from 0 to 5 of xsqrt(5-x) ?
The integral from 0 to 5 of xsqrt(5-x) is -10sqrt(5)+(50sqrt(5))/3