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Chapter 5: Problem 9

Symmetry in integrals Use symmetry to evaluate the following integrals. $$\int_{-2}^{2}\left(3 x^{8}-2\right) d x$$

Short Answer

Expert verified
Question: Evaluate the following definite integral using symmetry properties: $$\int_{-2}^{2}\left(3 x^{8}-2\right) d x$$ Answer: The definite integral evaluates to: $$\int_{-2}^{2}\left(3 x^{8}-2\right) d x = \frac{2992}{9}$$

Step by step solution

01

Identify the symmetry properties of the function

First, we will identify whether the given function, \(f(x)=3x^8 - 2\), has any symmetry properties. We will check for: 1. Even function symmetry: if \(f(-x) = f(x)\) 2. Odd function symmetry: if \(f(-x) = -f(x)\) Let's calculate \(f(-x)\): $$f(-x) = 3(-x)^8 - 2 = 3x^8 - 2$$ We can see that \(f(-x) = f(x)\), which means the given function is even.
02

Evaluate the integral for an even function

Now that we know the given function is even, we can use the symmetry property of even functions to evaluate the definite integral. For an even function, the integral can be expressed as: $$\int_{-a}^{a}f(x) dx = 2\int_{0}^{a}f(x) dx$$ So in our case, the definite integral becomes: $$\int_{-2}^{2}\left(3 x^{8}-2\right) d x = 2\int_{0}^{2}\left(3 x^{8}-2\right) d x$$
03

Evaluate the remaining integral

Now, we need to evaluate the remaining integral: $$2\int_{0}^{2}\left(3 x^{8}-2\right) d x$$ To do this, we will integrate each term and then evaluate them at the limits: $$2\left[\int_0^2 (3x^8)dx - \int_0^2 (2)dx\right]$$
04

Integrate the individual terms

Integrate the terms separately: $$2\left[\frac{3x^9}{9}\Big|_0^2 - 2x\Big|_0^2\right]$$
05

Evaluate the integrals at the limits

Now we will find the values of integrals at the limits and subtract them: $$2\left[\left(\frac{3(2^9)}{9} - 2(2)\right) - \left(\frac{3(0^9)}{9} - 2(0)\right)\right]$$
06

Calculate the result

Simplify the expression and solve for the final answer: $$2\left[\left(\frac{3(512)}{9} - 4\right) - \left(0 - 0\right)\right] = 2\left[\left(\frac{1536}{9} - 4\right)\right]$$ $$= 2\left[\frac{1496}{9}\right] = \frac{2992}{9}$$ So the definite integral evaluates to: $$\int_{-2}^{2}\left(3 x^{8}-2\right) d x = \frac{2992}{9}$$

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