What is the integral from 1 to 3 of xsqrt(3-x) ?

The integral from 1 to 3 of xsqrt(3-x) is -(8sqrt(2))/5+4sqrt(2)

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integral from 1 to 3 of xsqrt(3-x)

\int_{1}^{3} x 3 - x d x

- \frac{8 2}{5} + 42

Decimal

3.39411 \dots

Solution steps

\int_{1}^{3} x 3 - x d x

Apply u-substitution

: \int_{2}^{0} - (- u + 3) u d u

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

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

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

Take the constant out:

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

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

Expand

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

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

\int_{0}^{2} u^{\frac{3}{2}} d u = \frac{8 2}{5}

\int_{0}^{2} 3 u d u = 42

= - (- (- \frac{8 2}{5} + 42))

Simplify

= - \frac{8 2}{5} + 42

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What is the integral from 1 to 3 of xsqrt(3-x) ?
The integral from 1 to 3 of xsqrt(3-x) is -(8sqrt(2))/5+4sqrt(2)