What is the integral from 5 to x of 2/75 (x-5) ?

The integral from 5 to x of 2/75 (x-5) is 1/75 (x^2-10x+25)

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integral from 5 to x of 2/75 (x-5)

\int_{5}^{x} \frac{2}{75} (x - 5) d x

\frac{1}{75} (x^{2} - 10 x + 25)

Solution steps

\int_{5}^{x} \frac{2}{75} (x - 5) d x

Take the constant out:

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

= \frac{2}{75} \cdot \int_{5}^{x} x - 5 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{2}{75} (\int_{5}^{x} x d x - \int_{5}^{x} 5 d x)

\int_{5}^{x} x d x = \frac{x ^{2} - 25}{2}

\int_{5}^{x} 5 d x = 5 x - 25

= \frac{2}{75} (\frac{x ^{2} - 25}{2} - (5 x - 25))

Simplify

\frac{2}{75} (\frac{x ^{2} - 25}{2} - (5 x - 25)) : \frac{1}{75} (x^{2} - 10 x + 25)

= \frac{1}{75} (x^{2} - 10 x + 25)

What is the integral from 5 to x of 2/75 (x-5) ?
The integral from 5 to x of 2/75 (x-5) is 1/75 (x^2-10x+25)