What is the integral from 0 to 20 of xsqrt(20-x) ?

The integral from 0 to 20 of xsqrt(20-x) is -320sqrt(5)+(1600sqrt(5))/3

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integral from 0 to 20 of xsqrt(20-x)

\int_{0}^{20} x 20 - x d x

- 3205 + \frac{1600 5}{3}

Decimal

477.02783 \dots

Solution steps

\int_{0}^{20} x 20 - x d x

Apply u-substitution

: \int_{20}^{0} - (- u + 20) u d u

= \int_{20}^{0} - (- u + 20) u d u

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

= - \int_{0}^{20} - (- u + 20) u d u

Take the constant out:

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

= - (- \int_{0}^{20} (- u + 20) u d u)

Expand

(- u + 20) u : - u^{\frac{3}{2}} + 20 u

= - (- \int_{0}^{20} - u^{\frac{3}{2}} + 20 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}^{20} u^{\frac{3}{2}} d u + \int_{0}^{20} 20 u d u))

\int_{0}^{20} u^{\frac{3}{2}} d u = 3205

\int_{0}^{20} 20 u d u = \frac{1600 5}{3}

= - (- (- 3205 + \frac{1600 5}{3}))

Simplify

= - 3205 + \frac{1600 5}{3}

\int \frac{e ^{x}}{4} d x

f (x) = \frac{2}{x}, (5, \frac{2}{5})

y^{^{''}} + 12 y^{^{'}} + 35 y = 0

x \to 0 lim (\frac{4 - x}{x ^{2}})

\frac{5 ^{x^{3}}}{4 x ^{2}}

What is the integral from 0 to 20 of xsqrt(20-x) ?
The integral from 0 to 20 of xsqrt(20-x) is -320sqrt(5)+(1600sqrt(5))/3