What is the integral from 1 to 5 of x/(sqrt(5-x)) ?

The integral from 1 to 5 of x/(sqrt(5-x)) is 44/3

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integral from 1 to 5 of x/(sqrt(5-x))

\int_{1}^{5} \frac{x}{5 - x} d x

\frac{44}{3}

Decimal

14.66666 \dots

Solution steps

\int_{1}^{5} \frac{x}{5 - x} d x

Apply u-substitution

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

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

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

Take the constant out:

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

= - (- 2 \cdot \int_{0}^{2} - u^{2} + 5 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

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

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

\int_{0}^{2} 5 d u = 10

= - (- 2 (- \frac{8}{3} + 10))

Simplify

- (- 2 (- \frac{8}{3} + 10)) : \frac{44}{3}

= \frac{44}{3}

l a pl a ce t r an s f or m \frac{9}{4} t^{2} e^{2 t} e^{3 t}

\frac{d}{d x} (x tan (2 x) + 7)

\int - \frac{1}{64} e^{- 4 x} d x

y^{^{'}} = e^{2 y} sin (3 x)

t a y l or sin (x), \frac{π}{4}

What is the integral from 1 to 5 of x/(sqrt(5-x)) ?
The integral from 1 to 5 of x/(sqrt(5-x)) is 44/3