If int x f(x) dx + frac{f(x)}{2} = 0, then f( | vedclass

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(A) Given the equation: $\int x f(x) dx + \frac{f(x)}{2} = 0$.Differentiating both sides with respect to $x$ using the Fundamental Theorem of Calculus

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$C e^{-x^2}$

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(A) Given the equation: $\int x f(x) dx + \frac{f(x)}{2} = 0$.Differentiating both sides with respect to $x$ using the Fundamental Theorem of Calculus:$x f(x) + \frac{1}{2} f'(x) = 0$.Rearranging the terms to separate variables:$\frac{f'(x)}{f(x)} = -2x$.Integrating both sides with respect to $x$:$\int \frac{f'(x)}{f(x)} dx = \int -2x dx$.$\ln|f(x)| = -x^2 + C_1$.Exponentiating both sides:$f(x) = e^{-x^2 + C_1} = e^{C_1} \cdot e^{-x^2}$.Letting $e^{C_1} = C$, we get $f(x) = C e^{-x^2}$.

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

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