To evaluate the integral

$\int_{-1}^{1} \left(3x^8 - 4\right) \, dx$

we can apply symmetry to simplify our calculations. Let's analyze the integrand to determine if it has any symmetry properties that make the integration easier.

Step 1: Check Symmetry of the Integrand

1. Even function: A function $f(x)$ is even if $f(x) = f(-x)$. The integral of an even function over a symmetric interval $[-a, a]$ can be simplified as: $\int_{-a}^{a} f(x) \, dx = 2 \int_{0}^{a} f(x) \, dx$

2. Odd function: A function $f(x)$ is odd if $f(x) = -f(-x)$. The integral of an odd function over a symmetric interval $[-a, a]$ is zero: $\int_{-a}^{a} f(x) \, dx = 0$

Step 2: Analyze Each Term

Let's analyze each term in the integrand separately:

1. Term $3x^8$: This term is an even function because $x^8$ is even (i.e., $x^8 = (-x)^8$). Therefore, $3x^8$ is also even.

2. Term $-4$: This term is a constant, which is also even.

Since both terms in the integrand are even functions, the integrand $3x^8 - 4$ is an even function.

Step 3: Simplify the Integral Using Symmetry

Since the integrand is even, we can evaluate the integral over $[-1, 1]$ by doubling the integral over $[0, 1]$:

$\int_{-1}^{1} \left(3x^8 - 4\right) \, dx = 2 \int_{0}^{1} \left(3x^8 - 4\right) \, dx$

Step 4: Evaluate the Integral

Now, let's compute the integral:

$2 \int_{0}^{1} \left(3x^8 - 4\right) \, dx = 2 \left( \int_{0}^{1} 3x^8 \, dx - \int_{0}^{1} 4 \, dx \right)$

Integral of $3x^8$:

$\int_{0}^{1} 3x^8 \, dx = 3 \int_{0}^{1} x^8 \, dx = 3 \cdot \frac{x^9}{9} \Big|_{0}^{1} = 3 \cdot \frac{1}{9} = \frac{1}{3}$

Integral of $4$:

$\int_{0}^{1} 4 \, dx = 4x \Big|_{0}^{1} = 4 \cdot 1 = 4$

Combine Results

Substitute these results back:

$2 \left( \frac{1}{3} - 4 \right) = 2 \left( \frac{1}{3} - \frac{12}{3} \right) = 2 \left( -\frac{11}{3} \right) = -\frac{22}{3}$

Final Answer

$\int_{-1}^{1} \left(3x^8 - 4\right) \, dx = -\frac{22}{3}$

Would you like more details on any step, or have any other questions?

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1. How can we use symmetry to evaluate integrals with different limits?
2. What are the conditions for an integrand to be even or odd?
3. How would the approach differ if the function were not symmetric?
4. What are some examples of odd functions where symmetry simplifies the integral?
5. How does constant scaling impact the result in symmetric integrals?

Tip: Remember, even functions integrated over symmetric intervals $[-a, a]$ allow you to halve the interval and double the result.