What is the integral from 0 to 3 of (-x^2+2x+3) ?

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

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integral from 0 to 3 of (-x^2+2x+3)

\int_{0}^{3} (- x^{2} + 2 x + 3) d x

9

Solution steps

\int_{0}^{3} - x^{2} + 2 x + 3 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}^{3} x^{2} d x + \int_{0}^{3} 2 x d x + \int_{0}^{3} 3 d x

\int_{0}^{3} x^{2} d x = 9

\int_{0}^{3} 2 x d x = 9

\int_{0}^{3} 3 d x = 9

= - 9 + 9 + 9

Simplify

= 9

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