What is the integral from 0 to 12 of xsqrt(12-x) ?

The integral from 0 to 12 of xsqrt(12-x) is -(576sqrt(3))/5+192sqrt(3)

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integral from 0 to 12 of xsqrt(12-x)

\int_{0}^{12} x 12 - x d x

- \frac{576 3}{5} + 1923

Decimal

133.02150 \dots

Solution steps

\int_{0}^{12} x 12 - x d x

Apply u-substitution

= \int_{12}^{0} - (- u + 12) u d u

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

= - \int_{0}^{12} - (- u + 12) u d u

Take the constant out:

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

= - (- \int_{0}^{12} (- u + 12) u d u)

Expand

(- u + 12) u : - u^{\frac{3}{2}} + 12 u

= - (- \int_{0}^{12} - u^{\frac{3}{2}} + 12 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}^{12} u^{\frac{3}{2}} d u + \int_{0}^{12} 12 u d u))

\int_{0}^{12} u^{\frac{3}{2}} d u = \frac{576 3}{5}

\int_{0}^{12} 12 u d u = 1923

= - (- (- \frac{576 3}{5} + 1923))

Simplify

= - \frac{576 3}{5} + 1923

What is the integral from 0 to 12 of xsqrt(12-x) ?
The integral from 0 to 12 of xsqrt(12-x) is -(576sqrt(3))/5+192sqrt(3)