As written, the equation is syntactically ambiguous... the integration variable can't have the same name as a free variable, or else you can't tell which $x$ is meant by each occurrence within the integrand. There are four possible disambiguations:
- $y(x) = \int_{0}^{\pi} \sin(t+y(t)) dt.$
- $y(x) = \int_{0}^{\pi} \sin(t+y(x)) dt = 2 \cos(y(x))$.
- $y(x) = \int_{0}^{\pi} \sin(x+y(t)) dt = \sin(x)\int_{0}^{\pi}\cos(y(t))dt + \cos(x)\int_{0}^{\pi}\sin(y(t))dt.$
- $y(x) = \int_{0}^{\pi} \sin(x+y(x)) dt = \pi\sin(x+y(x))$.
In the first case, $y(x)$ is a constant $K$ satisfying $K=\int_{0}^{\pi}\sin(t+K)dt=2\cos K$; the unique solution is $K=1.0298665...$.
In the second case, $y(x)$ must satisfy $y=2\cos(y)$ at each point independently; since this has a unique solution, the result is the same as the first case.
In the third case, $y(x)=A\cos(x)+B\sin(x)$, where $A$ and $B$ satisfy a coupled integral equation.
In the fourth case, $y(x)$ satisfies an implicit equation that depends on $x$ at each point independently.