Question

If \( f(x) = \frac{x^3 + 5}{\sqrt{12 + x}} \) and } \[ \int_{-5}^{5}f(x) \, dx = \int_{0}^{5} \left( f(x) + g(x) \right) \, dx, \text{ then } g(x) = \]

• A. \( \frac{5 - x^3}{\sqrt{12 - x}} \)
• B. \( -\frac{5 + x^3}{\sqrt{12 + x}} \)
• C. \( -\frac{x^3 + 5}{\sqrt{12 + x}} \)
• D. \( \frac{5 + x^3}{\sqrt{12 - x}} \)

Hint:

When dealing with integrals involving odd and even functions, remember that integrals of odd functions over symmetric limits can be simplified using their properties.

Answer 1

The Correct Option is A

We are given that: \[ \int_{-5}^{5} f(x) \, dx = \int_{0}^{5} \left( f(x) + g(x) \right) \, dx \] We can split the left-hand side of the equation into two parts as follows: \[ \int_{-5}^{5} f(x) \, dx = \int_{-5}^{0} f(x) \, dx + \int_{0}^{5} f(x) \, dx \] From the given equation, we have: \[ \int_{-5}^{0} f(x) \, dx = \int_{0}^{5} g(x) \, dx \] Now, using the property of even and odd functions, we know that the function \( f(x) \) is odd, as it contains terms like \( x^3 \). Thus, for odd functions, we have: \[ f(-x) = -f(x) \] The integral from \( -5 \) to \( 0 \) is the negative of the integral from \( 0 \) to \( 5 \), so: \[ \int_{-5}^{0} f(x) \, dx = - \int_{0}^{5} f(x) \, dx \] Hence, the equation becomes: \[ -\int_{0}^{5} f(x) \, dx = \int_{0}^{5} g(x) \, dx \] Now, comparing the expressions, we can deduce that \( g(x) \) is the negative of \( f(x) \), but with the square root term adjusted for \( 12 - x \) instead of \( 12 + x \). This leads to the function: \[ g(x) = \frac{5 - x^3}{\sqrt{12 - x}} \] Hence, the correct answer is option (1).

Answer 2

Step 1: Understand the Given Equation
We are given a function:
\[ f(x) = \frac{x^3 + 5}{\sqrt{12 + x}} \]
and the relation:
\[ \int_{-5}^{5} f(x)\, dx = \int_0^5 \left( f(x) + g(x) \right)\, dx \]
We are asked to find the function \( g(x) \).

Step 2: Use the Property of Definite Integrals
Split the integral on the left-hand side:
\[ \int_{-5}^{5} f(x)\, dx = \int_{-5}^{0} f(x)\, dx + \int_0^5 f(x)\, dx \]
Now substitute into the given equation:
\[ \int_{-5}^{0} f(x)\, dx + \int_0^5 f(x)\, dx = \int_0^5 f(x)\, dx + \int_0^5 g(x)\, dx \]
Subtract \( \int_0^5 f(x)\, dx \) from both sides:
\[ \int_{-5}^{0} f(x)\, dx = \int_0^5 g(x)\, dx \]
So,
\[ g(x) = f(-x) \quad \text{(after change of variable)} \]

Step 3: Compute \( f(-x) \)
\[ f(-x) = \frac{(-x)^3 + 5}{\sqrt{12 - x}} = \frac{-x^3 + 5}{\sqrt{12 - x}} = \frac{5 - x^3}{\sqrt{12 - x}} \]

Final Answer:
\[ \boxed{g(x) = \frac{5 - x^3}{\sqrt{12 - x}}} \]