Use symmetry to evaluate the following integral.
Integral from negative 1 to 1 left parenthesis 5 x squared minus 2 right parenthesis dx∫−115x2−2 dx
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Part 1
Integral from negative 1 to 1 left parenthesis 5 x squared minus 2 right parenthesis dx∫−115x2−2 dxequals=enter your response here  ​(Type an integer or a simplified​ fraction.)

To evaluate the integral 

\[
\int_{-1}^{1} \left( 5x^2 - 2 \right) dx
\]

using symmetry, let us analyze the terms in the integrand:

1. **Symmetry of the terms**:
   - \( 5x^2 \): This term is an **even function**, since \( (5x^2) = 5(-x)^2 \), which simplifies back to \( 5x^2 \).
   - \( -2 \): This term is also even, as it is constant and independent of \( x \).

Thus, the entire function \( 5x^2 - 2 \) is even.

2. **Property of integrals for even functions**:
   For an even function \( f(x) \), the integral over a symmetric interval \( [-a, a] \) can be written as:
   \[
   \int_{-a}^a f(x) \, dx = 2 \int_0^a f(x) \, dx
   \]

3. **Break the integral**:
   Applying the property above:
   \[
   \int_{-1}^1 (5x^2 - 2) dx = 2 \int_0^1 (5x^2 - 2) dx
   \]

4. **Evaluate the simplified integral**:
   Compute \( \int_0^1 (5x^2 - 2) dx \):
   \[
   \int_0^1 (5x^2 - 2) dx = \int_0^1 5x^2 dx - \int_0^1 2 dx
   \]

   - For \( \int_0^1 5x^2 dx \):
     \[
     \int_0^1 5x^2 dx = 5 \int_0^1 x^2 dx = 5 \left[ \frac{x^3}{3} \right]_0^1 = 5 \cdot \frac{1^3}{3} - 5 \cdot \frac{0^3}{3} = \frac{5}{3}
     \]

   - For \( \int_0^1 2 dx \):
     \[
     \int_0^1 2 dx = 2 \left[ x \right]_0^1 = 2(1 - 0) = 2
     \]

   Combine the results:
   \[
   \int_0^1 (5x^2 - 2) dx = \frac{5}{3} - 2 = \frac{5}{3} - \frac{6}{3} = -\frac{1}{3}
   \]

5. **Multiply by 2 for symmetry**:
   \[
   \int_{-1}^1 (5x^2 - 2) dx = 2 \cdot \left( -\frac{1}{3} \right) = -\frac{2}{3}
   \]

### Final Answer:
\[
\int_{-1}^1 \left( 5x^2 - 2 \right) dx = -\frac{2}{3}
\]

Would you like a more detailed step for any part?  
Here are 5 related questions:
1. How do you determine whether a function is even or odd?
2. What are the properties of integrals for odd functions over symmetric intervals?
3. How would this problem change if the limits were not symmetric (e.g., \( \int_{0}^2 (5x^2 - 2) dx \))?
4. Can we use substitution methods to solve this integral in an alternative way?
5. What happens if the integrand includes both even and odd functions?

**Tip:** For symmetric limits, always check if the function is even or odd to simplify calculations!

Use symmetry to evaluate the following integral. Integral from -1 to 1 (5x^2 - 2) dx.

Learn how to evaluate the integral ∫[−1,1](5x^2 − 2)dx using the symmetry properties of even functions. This step-by-step solution includes key concepts, such as identifying even functions and applying integral formulas. Ideal for Grade 12 or first-year college students.

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