# How do you evaluate the definite integral int(-2x+2)dx from [-2,3]?

Nov 3, 2016

I found: ${\int}_{-} {2}^{3} \left(- 2 x + 2\right) \mathrm{dx} = 5$

#### Explanation:

We can write the integral as:

$\int - 2 x \mathrm{dx} + \int 2 \mathrm{dx} = - 2 \int x \mathrm{dx} + 2 \int \mathrm{dx} =$

we can now integrate to get:

$= - 2 {x}^{2} / 2 + 2 x = - {x}^{2} + 2 x$

we now substitute the exteremes of integration into our result and subtract:

for $x = 3$ we get::$- 9 + 6 = - 3$

for $x = - 2$ we get: $- 4 - 4 = - 8$

Let us subtract them:
$- 3 - \left(- 8\right) = 5$
so finally:

${\int}_{-} {2}^{3} \left(- 2 x + 2\right) \mathrm{dx} = 5$