evaluate the definite integral:
$$ int _ { - 2 } ^ { 6 } ( 2 x - e ^ { x } ) d x = $$

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Accepted Answer

# Explanation:
## Step1: Find the antiderivative
The antiderivative of $2x$ is $x^{2}$ (using the power rule $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$, here $n = 1$), and the antiderivative of $-e^{x}$ is $-e^{x}$ (since $\int e^{x}dx=e^{x}+C$). So the antiderivative of $2x - e^{x}$ is $F(x)=x^{2}-e^{x}$.
## Step2: Apply the Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus states that $\int_{a}^{b}f(x)dx=F(b)-F(a)$. Here $a=-2$, $b = 6$.
$$
\begin{align*}
F(6)-F(-2)&=(6^{2}-e^{6})-((-2)^{2}-e^{-2})\
&=(36 - e^{6})-(4 - e^{-2})\
&=36 - e^{6}-4 + e^{-2}\
&=32 - e^{6}+\frac{1}{e^{2}}
\end{align*}
$$

# Answer:
$32 - e^{6}+\frac{1}{e^{2}}$

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QUESTION IMAGE

Question

Explanation:

Step1: Find the antiderivative

Step2: Apply the Fundamental Theorem of Calculus

Answer: