It looks like you've written an integral expression, but it's a bit unclear due to formatting issues. I'll attempt to break it down based on what I understand:

It seems you're trying to express the following:

$\int_{-1}^{1} \left( 1 + x + 1 + x - (1 + 2x + 2x^2 - 2x + 2x^2) \right) \, dx$

This seems to be an integral with limits from $-1$ to $1$, and the integrand is a function involving terms in $x$.

Let's simplify the expression inside the integral:

$1 + x + 1 + x - (1 + 2x + 2x^2 - 2x + 2x^2)$

Simplify the terms inside the parentheses:

$1 + x + 1 + x - (1 + 2x + 2x^2 - 2x + 2x^2)$

$2 + 2x - (1 + 2x + 2x^2 - 2x + 2x^2)$

Now distribute the negative sign:

$2 + 2x - 1 - 2x - 2x^2 + 2x - 2x^2$

Now simplify:

$(2 - 1) + (2x - 2x + 2x) - (2x^2 + 2x^2)$

$1 + 2x - 4x^2$

Thus, the integral becomes:

$\int_{-1}^{1} (1 + 2x - 4x^2) \, dx$

Now we can integrate each term individually.

• The integral of $1$ with respect to $x$ is $x$.
• The integral of $2x$ with respect to $x$ is $x^2$.
• The integral of $-4x^2$ with respect to $x$ is $-\frac{4x^3}{3}$.

So, the integral becomes:

$\left[ x + x^2 - \frac{4x^3}{3} \right]_{-1}^{1}$

Now, evaluate this expression at the limits:

At $x = 1$:

$1 + 1^2 - \frac{4(1)^3}{3} = 1 + 1 - \frac{4}{3} = 2 - \frac{4}{3} = \frac{6}{3} - \frac{4}{3} = \frac{2}{3}$

At $x = -1$:

$-1 + (-1)^2 - \frac{4(-1)^3}{3} = -1 + 1 - \frac{-4}{3} = 0 + \frac{4}{3} = \frac{4}{3}$

Now subtract the value at $x = -1$ from the value at $x = 1$:

$\frac{2}{3} - \frac{4}{3} = -\frac{2}{3}$

Thus, the value of the integral is:

$\boxed{-\frac{2}{3}}$

Would you like further clarification or more details on any part of the solution?

Related Questions:

1. How does the symmetry of the function affect the integral?
2. Can you calculate the definite integral for other polynomial functions?
3. What happens if we change the limits of the integral?
4. How does the presence of even and odd functions influence the integral's result?
5. What would the result be if we added a constant to the function being integrated?

Tip:

If you're unsure about the steps in simplifying the integral, try breaking down the expression into smaller parts and simplifying each term before performing the integration.