How do you evaluate definite integral $\int {\left( { - 2x + 2} \right)dx} $ from $\left[ { - 2,3} \right]$ ?

Hint: In this question, we are given an integral and we have been asked to integrate it and we have also been given a range, within which we have to find the definite integral. So, first integrate the given terms and then put the upper limit and from it, subtract the lower limit. You will get your answer.

Formulas used: $\int {{x^n}dx = \dfrac{{{x^{n + 1}}}}{{n + 1}} + C} $
$\int {dx = x + C} $

Complete step by step answer:
We are being given an integral and we have to integrate it. Let’s see how it is done.
$ \Rightarrow \int {\left( { - 2x + 2} \right)dx} $ … (given)
Along with this, we are also given a range within which we have to calculate the integral.
We can write it as –
$ \Rightarrow \int\limits_{ - 2}^3 {( - 2x + 2)dx} $
Now, using two formulas, we will integrate the given integral.
$ \Rightarrow \int {{x^n}dx = \dfrac{{{x^{n + 1}}}}{{n + 1}} + C} $ , and
$ \Rightarrow \int {dx = x + C} $
Now, integrating the given integral
$ \Rightarrow \left[ {\dfrac{{ - 2{x^{1 + 1}}}}{{1 + 1}} + 2x} \right]_{ - 2}^3$
Simplifying it further,
$ \Rightarrow \left[ {\dfrac{{ - 2{x^2}}}{2} + 2x} \right]_{ - 2}^3$
$ \Rightarrow \left[ { - {x^2} + 2x} \right]_{ - 2}^3$
Now, we will put the upper limit first and then from it, we will subtract the lower limit.
Putting $x = 3$ in the answer,
$ \Rightarrow - {\left( 3 \right)^2} + 2\left( 3 \right)$
On simplifying, we get,
$ \Rightarrow - 9 + 6 = - 3$ …. (1)
Now, we will put $x = - 2$ in the answer,
$ \Rightarrow - {\left( { - 2} \right)^2} + 2\left( { - 2} \right)$
On simplifying, we will get,
$ \Rightarrow - 4 - 4 = - 8$ …. (2)
Subtracting (2) from (1),
$ \Rightarrow - 3 - \left( { - 8} \right) = 5$

Hence, our answer is $5$.

Note: What is Integration?
Integration is known as – ‘antiderivative’. It is the process of finding the function, whose derivative is given.
If we are given the following integration, how will we find the answer?
$ \Rightarrow \int\limits_a^b {f'\left( x \right)} dx$
In definite integral, we put the upper limit and the lower limit like below:
$ \Rightarrow f\left( b \right) - f\left( a \right)$
It is to be noted that we do not add a constant of integration in definite integral as we are already dealing with the constants in definite integral, unlike in indefinite integral.