What is the integral from-1 to 3 of (-x^2+3x)-(x-3) ?

The integral from-1 to 3 of (-x^2+3x)-(x-3) is 32/3

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integral from-1 to 3 of (-x^2+3x)-(x-3)

\int_{- 1}^{3} (- x^{2} + 3 x) - (x - 3) d x

\frac{32}{3}

Solution steps

\int_{- 1}^{3} - x^{2} + 3 x - (x - 3) d x

Simplify

= \int_{- 1}^{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_{- 1}^{3} x^{2} d x + \int_{- 1}^{3} 2 x d x + \int_{- 1}^{3} 3 d x

\int_{- 1}^{3} x^{2} d x = \frac{28}{3}

\int_{- 1}^{3} 2 x d x = 8

\int_{- 1}^{3} 3 d x = 12

= - \frac{28}{3} + 8 + 12

Simplify

- \frac{28}{3} + 8 + 12 : \frac{32}{3}

= \frac{32}{3}

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What is the integral from-1 to 3 of (-x^2+3x)-(x-3) ?
The integral from-1 to 3 of (-x^2+3x)-(x-3) is 32/3