I've been given to evaluate these integrals using orthogonal polynomials method. But I hardly can find any info about this method, so I'm asking for any guides or theory to solve these two integrals.
$\int_{-1}^1\phi(\xi)[{\ln \frac{1}{|x-\xi|}+\frac{e^{-(x^2+1)}}{(\xi-2)}}]d\xi=\sin(x), |x|<1$
$\frac{d^2}{dx^2}\int_{-1}^1\phi(\xi)\ln\frac1{|x-\xi|}+\int_{-1}^1\frac{e^{-(x+\xi)}}{\sqrt{x^2+2}}\phi(\xi)d\xi=(x^2+1)\cdot2^{-x}, |x|<1$