What is the integral from 0 to 4 of (-x^2+2x+1) ?

The integral from 0 to 4 of (-x^2+2x+1) is -4/3

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integral from 0 to 4 of (-x^2+2x+1)

\int_{0}^{4} (- x^{2} + 2 x + 1) d x

- \frac{4}{3}

Solution steps

\int_{0}^{4} - x^{2} + 2 x + 1 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}^{4} x^{2} d x + \int_{0}^{4} 2 x d x + \int_{0}^{4} 1 d x

\int_{0}^{4} x^{2} d x = \frac{64}{3}

\int_{0}^{4} 2 x d x = 16

\int_{0}^{4} 1 d x = 4

= - \frac{64}{3} + 16 + 4

Simplify

- \frac{64}{3} + 16 + 4 : - \frac{4}{3}

= - \frac{4}{3}

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

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\int_{0}^{5} (x^{2} - x + 1) e^{2 x} d x

What is the integral from 0 to 4 of (-x^2+2x+1) ?
The integral from 0 to 4 of (-x^2+2x+1) is -4/3