Evaluate the following.∫(−2𝑥)d𝑥=

Sure, let's evaluate the integral step by step.

Given the integral: $\int (-2x) \, dx$

Step 1: Identify the integrand. The integrand is $-2x$.

Step 2: Apply the power rule for integration. The power rule for integration states that: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ where $n \neq -1$ and $C$ is the constant of integration.

Step 3: Integrate $-2x$. Here, $-2x$ can be written as $-2 \cdot x^1$. Using the power rule: $\int (-2x) \, dx = -2 \int x \, dx$

Step 4: Integrate $x$. $\int x \, dx = \int x^1 \, dx = \frac{x^{1+1}}{1+1} = \frac{x^2}{2}$

Step 5: Multiply by the constant $-2$. $-2 \cdot \frac{x^2}{2} = -x^2$

Step 6: Add the constant of integration $C$. $-x^2 + C$

So, the evaluated integral is: $\int (-2x) \, dx = -x^2 + C$