# 2010-11 Integral Equation

Let z be a real number. Find all solutions of the following integral equation: $$f(x)=e^x+z \int_0^1 e^{x-y} f(y)\,dy$$ for 0≤x≤1.

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$$\displaystyle \lim_{\varepsilon\to 0}\int_0^{2\varepsilon} \log\left(\frac{|\sin t-\varepsilon|}{\sin \varepsilon}\right) \frac{dt}{\sin t}$$.