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

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Answer 1 · 2016-11-03T19:20:16

I found: $\int_{- 2}^{3} (- 2 x + 2) d x = 5$ $int_-2^3(-2x+2)dx=5$

We can write the integral as:

$\int - 2 x d x + \int 2 d x = - 2 \int x d x + 2 \int d x =$ $int-2xdx+int2dx=-2intxdx+2intdx=$

we can now integrate to get:

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

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

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

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

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

$\int_{- 2}^{3} (- 2 x + 2) d x = 5$ $int_-2^3(-2x+2)dx=5$

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