Discussion
The following integral equation
\[ x(t)=\sin t+\lambda\int_{0}^{2\pi}\sin(t+s)\,x(s)\,ds \]
is :
Answer: A
It is a Fredholm integral equation of the second kind.
The limits of integration are fixed (0 to \(2\pi\)), which makes it Fredholm; a Volterra equation has a variable upper limit such as t. The unknown x(t) also appears outside the integral, which makes it of the second kind:
\[ x(t) = f(t) + \lambda\int_{a}^{b} K(t, s)\, x(s)\, ds \]
In an equation of the first kind the unknown appears only under the integral sign, \(f(t) = \lambda\int K(t, s)x(s)\, ds\).
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