I'm a beginner in the Fourier series and I can't find the solution to the below integral and relationship with the Fourier series. If
$$f(x)=1+\sum^\infty_{n=1}\frac{(-1)^n}{n}\cos(nx)+\frac{1}{n^2}\sin(nx)$$
calculate
$$I=\int^\pi_{-\pi}f(x)(\sin(2.5x)+\cos(2.5x))^2\cos(5x)dx.$$
Any help and hint are much appreciated.