Understanding the Definite Integral Value

The question asks us to evaluate the definite integral:
$$ \rm \int_{-2}^{\ 2}(ax^5 + bx^3 + c)\ dx $$ and determine which coefficient (\(a\), \(b\), or \(c\)) influences its value. We will use the properties of definite integrals and functions with symmetric limits.

Properties of Definite Integrals with Symmetric Limits

A key property for definite integrals with symmetric limits, like from \(-a\) to \(a\), relates to the integrand's symmetry:

• If the integrand \(f(x)\) is an odd function (meaning \(f(-x) = -f(x)\)), then the integral over symmetric limits is zero: \(\rm \int_{-a}^{a} f(x)\ dx = 0\).
• If the integrand \(f(x)\) is an even function (meaning \(f(-x) = f(x)\)), then the integral over symmetric limits can be simplified: \(\rm \int_{-a}^{a} f(x)\ dx = 2 \int_{0}^{a} f(x)\ dx\).

Analyzing the Integrand

Let the integrand be \(f(x) = ax^5 + bx^3 + c\). We can analyze each term separately:

1. Term 1: \(ax^5\)

   Let \(g(x) = ax^5\). We check if it's odd or even:

   \(g(-x) = a(-x)^5 = a(-x^5) = -ax^5 = -g(x)\)

   Since \(g(-x) = -g(x)\), the term \(ax^5\) represents an odd function.

   Therefore, its integral over the symmetric limits \(-2\) to \(2\) is zero:

   $$ \rm \int_{-2}^{\ 2} ax^5\ dx = 0 $$

   This means the value of \(a\) does not affect the value of this part of the integral.

2. Term 2: \(bx^3\)

   Let \(h(x) = bx^3\). We check its symmetry:

   \(h(-x) = b(-x)^3 = b(-x^3) = -bx^3 = -h(x)\)

   Since \(h(-x) = -h(x)\), the term \(bx^3\) is also an odd function.

   Its integral over the symmetric limits \(-2\) to \(2\) is also zero:

   $$ \rm \int_{-2}^{\ 2} bx^3\ dx = 0 $$

   This means the value of \(b\) does not affect the value of this part of the integral.

3. Term 3: \(c\)

   Let \(k(x) = c\) (a constant term). We check its symmetry:

   \(k(-x) = c = k(x)\)

   Since \(k(-x) = k(x)\), the term \(c\) is an even function.

   Its integral over the symmetric limits \(-2\) to \(2\) is calculated as:

   $$ \rm \int_{-2}^{\ 2} c\ dx = 2 \int_{0}^{2} c\ dx $$

   Evaluating this integral:

   $$ 2 \int_{0}^{2} c\ dx = 2 \left[ cx \right]_{0}^{2} = 2 (c(2) - c(0)) = 2(2c) = 4c $$

   The value of this part of the integral is \(4c\). This value clearly depends on the value of \(c\).

Conclusion on Integral Value

Combining the results for all terms:

$$ \rm \int_{-2}^{\ 2}(ax^5 + bx^3 + c)\ dx = \int_{-2}^{\ 2} ax^5\ dx + \int_{-2}^{\ 2} bx^3\ dx + \int_{-2}^{\ 2} c\ dx $$

$$ = 0 + 0 + 4c = 4c $$

The final value of the definite integral is \(4c\). This demonstrates that the value of the integral depends solely on the coefficient \(c\).