$$ \int _{}^{}f(x)+x\frac{d}{dx}(f(x)) d x=0 $$

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To solve the integral equation

$$
\int \left( f(x) + x \frac{d}{dx}(f(x)) \right) dx = 0,
$$

we can start by simplifying the integrand. The term $$ x \frac{d}{dx}(f(x)) $$ can be rewritten using the product rule of differentiation. Specifically, we can express it as:

$$
x \frac{d}{dx}(f(x)) = \frac{d}{dx}(x f(x)) - f(x).
$$

This allows us to rewrite the integrand:

$$
f(x) + x \frac{d}{dx}(f(x)) = f(x) + \left( \frac{d}{dx}(x f(x)) - f(x) \right) = \frac{d}{dx}(x f(x)).
$$

Thus, the integral becomes:

$$
\int \frac{d}{dx}(x f(x)) \, dx = x f(x) + C,
$$

where $$ C $$ is the constant of integration.

Setting this equal to zero gives us:

$$
x f(x) + C = 0.
$$

From this, we can solve for $$ f(x) $$:

$$
x f(x) = -C \quad \Rightarrow \quad f(x) = -\frac{C}{x}.
$$

Thus, the general solution for $$ f(x) $$ is:

$$
f(x) = -\frac{C}{x},
$$

where $$ C $$ is an arbitrary constant.
The general solution for $$ f(x) $$ is $$ f(x) = -\frac{C}{x} $$, where $$ C $$ is an arbitrary constant.

\( \int _{}^{}f(x)+x\frac{d}{dx}(f(x)) d x=0 \)

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