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

The integral from-4 to 1 of (-x+4)-(x^2+2x) is 125/6

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

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

\frac{125}{6}

Decimal

20.83333 \dots

Solution steps

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

Simplify

= \int_{- 4}^{1} - x^{2} - 3 x + 4 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_{- 4}^{1} x^{2} d x - \int_{- 4}^{1} 3 x d x + \int_{- 4}^{1} 4 d x

\int_{- 4}^{1} x^{2} d x = \frac{65}{3}

\int_{- 4}^{1} 3 x d x = - \frac{45}{2}

\int_{- 4}^{1} 4 d x = 20

= - \frac{65}{3} - (- \frac{45}{2}) + 20

Simplify

- \frac{65}{3} - (- \frac{45}{2}) + 20 : \frac{125}{6}

= \frac{125}{6}

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\int_{1}^{5} (- x^{2} + 3 x + 2) d x

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What is the integral from-4 to 1 of (-x+4)-(x^2+2x) ?
The integral from-4 to 1 of (-x+4)-(x^2+2x) is 125/6