What is the integral from 0 to 5 of (5x-x^2) ?

The integral from 0 to 5 of (5x-x^2) is 125/6

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integral from 0 to 5 of (5x-x^2)

\int_{0}^{5} (5 x - x^{2}) d x

\frac{125}{6}

Decimal

20.83333 \dots

Solution steps

\int_{0}^{5} 5 x - x^{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

= \int_{0}^{5} 5 x d x - \int_{0}^{5} x^{2} d x

\int_{0}^{5} 5 x d x = \frac{125}{2}

\int_{0}^{5} x^{2} d x = \frac{125}{3}

= \frac{125}{2} - \frac{125}{3}

Simplify

\frac{125}{2} - \frac{125}{3} : \frac{125}{6}

= \frac{125}{6}

\int 9 d x

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

\frac{4 x}{x + 1}, at (3.3)

\int \frac{1}{x ^{5}} d x

x \to - 4 lim (\frac{5}{x + 4})

What is the integral from 0 to 5 of (5x-x^2) ?
The integral from 0 to 5 of (5x-x^2) is 125/6