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Answer the following True or False. $\newline$ $\int_{-a}^{a}(3x^{3}+9x)dx=0$ $\newline$ True $\newline$ False

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Properties of matrices

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Q. Answer the following True or False.

\newline

\int_{-a}^{a}(3x^{3}+9x)dx=0

\newline

True

\newline

False

Function Analysis: Let's consider the function $f(x) = 3x^3 + 9x$ . We need to determine if the integral of $f(x)$ from $-a$ to $a$ is zero. This function is an odd function because $f(-x) = -f(x)$ for all $x$ in the domain of $f$ . This is because the powers of $x$ are odd ( $3$ and $1$ ), and the coefficients are constants. Odd functions have the property that their integral over a symmetric interval about the origin is zero.
Integral Setup: Now, let's set up the integral: $\int_{-a}^{a} (3x^3 + 9x) \, dx$ . Since we have established that the function is odd, we can apply the property of odd functions to integrals. The integral of an odd function over a symmetric interval $[-a, a]$ is $0$ .
Conclusion: We can conclude that $\int_{-a}^{a} (3x^3 + 9x) \, dx = 0$ , because the function $3x^3 + 9x$ is an odd function and the interval is symmetric about the origin.

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