What is the integral from 0 to 21 of xsqrt(21-x) ?

The integral from 0 to 21 of xsqrt(21-x) is -(882sqrt(21))/5+294sqrt(21)

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integral from 0 to 21 of xsqrt(21-x)

\int_{0}^{21} x 21 - x d x

- \frac{882 21}{5} + 29421

Decimal Notation

538.91090 \dots

Solution steps

\int_{0}^{21} x 21 - x d x

Apply u-substitution

: \int_{21}^{0} - (- u + 21) u d u

= \int_{21}^{0} - (- u + 21) u d u

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

= - \int_{0}^{21} - (- u + 21) u d u

Take the constant out:

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

= - (- \int_{0}^{21} (- u + 21) u d u)

Expand

(- u + 21) u : - u^{\frac{3}{2}} + 21 u

= - (- \int_{0}^{21} - u^{\frac{3}{2}} + 21 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}^{21} u^{\frac{3}{2}} d u + \int_{0}^{21} 21 u d u))

\int_{0}^{21} u^{\frac{3}{2}} d u = \frac{882 21}{5}

\int_{0}^{21} 21 u d u = 29421

= - (- (- \frac{882 21}{5} + 29421))

Simplify

= - \frac{882 21}{5} + 29421

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What is the integral from 0 to 21 of xsqrt(21-x) ?
The integral from 0 to 21 of xsqrt(21-x) is -(882sqrt(21))/5+294sqrt(21)