What is the integral from 0 to 5 of 1/4 (5-x)sqrt(x) ?

The integral from 0 to 5 of 1/4 (5-x)sqrt(x) is (5sqrt(5))/3

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integral from 0 to 5 of 1/4 (5-x)sqrt(x)

\int_{0}^{5} \frac{1}{4} (5 - x) x d x

\frac{5 5}{3}

Decimal Notation

3.72677 \dots

Solution steps

\int_{0}^{5} \frac{1}{4} (5 - x) x d x

Take the constant out:

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

= \frac{1}{4} \cdot \int_{0}^{5} (5 - x) x d x

Expand

(5 - x) x : 5 x - x^{\frac{3}{2}}

= \frac{1}{4} \cdot \int_{0}^{5} 5 x - x^{\frac{3}{2}} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{4} (\int_{0}^{5} 5 x d x - \int_{0}^{5} x^{\frac{3}{2}} d x)

\int_{0}^{5} 5 x d x = \frac{50 5}{3}

\int_{0}^{5} x^{\frac{3}{2}} d x = 105

= \frac{1}{4} (\frac{50 5}{3} - 105)

Simplify

\frac{1}{4} (\frac{50 5}{3} - 105) : \frac{5 5}{3}

= \frac{5 5}{3}

\frac{d}{d t} (v (t) t)

\frac{\partial}{\partial x} (x^{2} - 2 x y)

\frac{\partial}{\partial x} (y^{2} x)

\int \frac{4 + x ^{2}}{x} d x

x \to - 3 lim (x^{2} - 9)

What is the integral from 0 to 5 of 1/4 (5-x)sqrt(x) ?
The integral from 0 to 5 of 1/4 (5-x)sqrt(x) is (5sqrt(5))/3