Calculate integrals using numpy.fft

Question

Good evening,

I would like to understand why I do not get the correct result: I assume that I know my function on discrete data points and expand it as a discrete Fourier transform: $\text{sin}(x)=\sum_{k=-m}^m A_ke^{i2\pi kx}$

This I use to calculate an integral: $$\int_{-R}^{R}\text{sin}(x+y)\text{d}x = \sum_{k=-m}^m A_ke^{i2\pi ky}\int_{-R}^R e^{i2\pi kx} \text{d}x = \sum_{k=-m}^m\frac{A_k}{i2\pi k}\left( e^{i2\pi kR}-e^{-i2\pi kR} \right) e^{i2\pi ky}$$

In other words, I calculate $A_k$ from FFT(sin(x)), multiply each $A_k$ with the term $\frac{1}{i2\pi k}\left( e^{i2\pi kR}-e^{-i2\pi kR} \right) $ and go back into real space with inverse FFT of these modified $A_k$.

Yet I do not get the correct result (the analyitc solution is $-\text{cos}(R+y)+\text{cos}(-R+y)$ ). What am I doing wrong?

import numpy as np
import matplotlib.pyplot as plt
from numpy.fft import fft, fftfreq, ifft

pi2 = 2*np.pi

L = pi2
N = 1001   ## should be odd!!
R = 0.2

x = np.linspace(0, L-L/N, N)
f = np.sin(x)
FOU_f = np.zeros(N, dtype=complex)
phi = np.zeros(N, dtype=complex)
FOU_phi = np.zeros(N, dtype=complex)

FOU_f = fft(f, N)    ## function to fourier space

for k in range(1, N):
    FOU_phi[k] = ( FOU_f[k]/(1j*pi2*k) ) * ( np.exp(1j*pi2*k*R) - np.exp(-1j*pi2*k*R) )

phi = ifft(FOU_phi, N)  ## back to real space

analytic_sol = -np.cos(x+R) + np.cos(-R+x)

plt.plot(x, phi, label="test")
plt.plot(x, analytic_sol, label="analytic")
plt.legend()
plt.show()