IE16

Solve: $g(s) = f(s) + \lambda \int_0^{2\pi} \sin(s) \cos(t) g(t) dt \,$

$g(s) = f(s) + \lambda \sin(s) \int_0^{2\pi} \cos(t) g(t) dt \,$

Let $c = \int_0^{2\pi} \cos(t) g(t) dt\,$ .

Write a new equation:

$c = \int_0^{2\pi} \cos(t) \left[ f(t) + \lambda \sin(t) c \right] dt \,$

$c = \int_0^{2\pi} \cos(t) f(t) dt + \lambda c \int_0^{2\pi} \cos(t) \sin(t) dt\,$

The last integrand is odd and the limits of integration are a multiple of its period, so the integral equals 0.

$c = \int_0^{2\pi} \cos(t) f(t) dt\,$

From

$g(s) = f(s) + \lambda \sin(s) c \,$

the answer is

$g(s) = f(s) + \lambda \sin(s) \int_0^{2\pi} \cos(t) f(t) dt \,$