Question

$\int x f(x) dx + \frac{f(x)}{2} = 0$, then $f(x)$ is equal to

• A. $e^{-2x}$
• B. $e^{2x}$
• C. $e^{-x^2}$
• D. $e^{x^2}$

Hint:

Whenever you see an equation containing an integral of an unknown function (an integral equation), the standard first step is almost always to differentiate the entire equation with respect to the variable. This converts it into a manageable differential equation.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We are given an integral equation involving an unknown function $f(x)$. To solve for $f(x)$, we need to eliminate the integral sign. This is done by differentiating the entire equation with respect to $x$.

Step 2: Key Formula or Approach:
1. Differentiate both sides of the given equation $\int x f(x) dx + \frac{f(x)}{2} = 0$ with respect to $x$. Remember that $\frac{d}{dx} \int g(x) dx = g(x)$. 2. This will yield a first-order differential equation for $f(x)$. 3. Solve the differential equation using separation of variables to find the form of $f(x)$.

Step 3: Detailed Explanation:
The given equation is: \[ \int x f(x) dx + \frac{f(x)}{2} = 0 \] Let's differentiate both sides with respect to $x$: \[ \frac{d}{dx} \left[ \int x f(x) dx \right] + \frac{d}{dx} \left[ \frac{f(x)}{2} \right] = \frac{d}{dx}(0) \] Applying the fundamental theorem of calculus, the derivative of the integral is just the integrand: \[ x f(x) + \frac{1}{2} f'(x) = 0 \] Now we have a differential equation. Let's write $f'(x)$ as $\frac{dy}{dx}$ and $f(x)$ as $y$ to make it clearer: \[ x y + \frac{1}{2} \frac{dy}{dx} = 0 \] Rearrange to separate variables: \[ \frac{1}{2} \frac{dy}{dx} = -xy \] \[ \frac{dy}{dx} = -2xy \] Separate the variables $y$ and $x$: \[ \frac{1}{y} dy = -2x dx \] Integrate both sides: \[ \int \frac{1}{y} dy = \int -2x dx \] \[ \ln|y| = -x^2 + C \] To find $y$ (which is $f(x)$), exponentiate both sides: \[ y = e^{-x^2 + C} = e^C \cdot e^{-x^2} \] Let $e^C$ be a new constant $K$. So, the general form of the function is: \[ f(x) = K e^{-x^2} \] Looking at the given options, they are all specific cases where the constant $K$ is assumed to be $1$. The option that matches the functional form $e^{-x^2}$ is option (3).

Step 4: Final Answer:
The function $f(x)$ is $e^{-x^2}$.