What is the integral from 0 to 2 of ((-x+5)-(x^2-9)) ?

The integral from 0 to 2 of ((-x+5)-(x^2-9)) is 70/3

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integral from 0 to 2 of ((-x+5)-(x^2-9))

\int_{0}^{2} ((- x + 5) - (x^{2} - 9)) d x

\frac{70}{3}

Decimal

23.33333 \dots

Solution steps

\int_{0}^{2} - x + 5 - (x^{2} - 9) d x

Simplify

= \int_{0}^{2} - x^{2} - x + 14 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}^{2} x^{2} d x - \int_{0}^{2} x d x + \int_{0}^{2} 14 d x

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

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

\int_{0}^{2} 14 d x = 28

= - \frac{8}{3} - 2 + 28

Simplify

- \frac{8}{3} - 2 + 28 : \frac{70}{3}

= \frac{70}{3}

What is the integral from 0 to 2 of ((-x+5)-(x^2-9)) ?
The integral from 0 to 2 of ((-x+5)-(x^2-9)) is 70/3