La integral que debemos hallar es <img src="https://tex.z-dn.net/?f=%5Cdisplaystyle%0A%5Cint_%7B-25%7D%5E%7B25%7D%5Csqrt%7B1%2B%5Cpi%5E2%5Csin%5E2%5Cdfrac%7B%5Cpi%20x%7D%7B50%7D%7D%5C%3Bdx" />
si dividimos al intervalo [ - 25, 25] en 10 partes, entonces tendremos <img src="https://tex.z-dn.net/?f=x_k%3D-25%2B5k%5C%3B%2C%5C%3B%20k%5Cin%5B0%2C10%5D" />
por ende <img src="https://tex.z-dn.net/?f=f%28x_k%29%3D%5Csqrt%7B1%2B%5Cpi%5E2%5Csin%5E2%5Cdfrac%7B%5Cpi%20k%7D%7B5%7D%7D" />
Según la regla de Simpson tenemos que la integral mencionada arriba, se aproxima a :
<img src="https://tex.z-dn.net/?f=I%3D%5Cdfrac%7B25-%28-25%29%7D%7B3%2810%29%7D%5Bf%28x_0%29%2B4f%28x_1%29%2B2f%28x_2%29%2B4f%28x_3%29%2B2f%28x_4%29%2B4f%28x_5%29%2B%5Ccdots%5C%5C%20%5C%5C%0A%5Ccdots%20%2B%202f%28x_6%29%2B4f%28x_7%29%2B2f%28x_8%29%2B4f%28x_9%29%2Bf%28x_%7B10%7D%29%5D%5C%5C%20%5C%5C%20%5C%5C%0A%0A" /> <img src="https://tex.z-dn.net/?f=f%28x_0%29%3D1%5C%5C%20%5C%5C%0Af%28x_1%29%3D%5Csqrt%7B1%2B%5Cpi%5E2%5Csin%5E2%5Cdfrac%7B%5Cpi%7D%7B5%7D%7D%5C%5C%20%5C%5C%0Af%28x_2%29%3D%5Csqrt%7B1%2B%5Cpi%5E2%5Csin%5E2%5Cdfrac%7B2%5Cpi%7D%7B5%7D%7D%5C%5C%20%5C%5C%0Af%28x_3%29%3D%5Csqrt%7B1%2B%5Cpi%5E2%5Csin%5E2%5Cdfrac%7B3%5Cpi%7D%7B5%7D%7D%5C%5C%20%5C%5C%0Af%28x_4%29%3D%5Csqrt%7B1%2B%5Cpi%5E2%5Csin%5E2%5Cdfrac%7B4%5Cpi%7D%7B5%7D%7D%5C%5C%20%5C%5C%0Af%28x_5%29%3D1%5C%5C%20%5C%5C%0Af%28x_6%29%3Df%28x_1%29%5C%5C%20%5C%5C%0Af%28x_7%29%3Df%28x_2%29%5C%3B%2C%5C%3Bf%28x_8%29%3Df%28x_3%29%5C%3B%2C%5C%3B%20f%28x_9%29%3Df%28x_4%29%5C%3B%2C%5C%3Bf%28x_%7B10%7D%29%3D1%0A" />
<img src="https://tex.z-dn.net/?f=%5Cdisplaystyle%0AI%3D%5Cdfrac%7B5%7D%7B3%7D%5Ccdot6%5Csum_%7Bk%3D0%7D%5E%7B4%7D%5Csqrt%7B1%2B%5Cpi%5E2%5Csin%5E2%5Cdfrac%7Bk%5C%2C%5Cpi%7D%7B5%7D%7D%5Cright%5D%5C%5C%20%5C%5C%20%5C%5C%0A%5Cboxed%7BI%3D10%5Csum_%7Bk%3D0%7D%5E%7B4%7D%5Csqrt%7B1%2B%5Cpi%5E2%5Csin%5E2%5Cdfrac%7Bk%5C%2C%5Cpi%7D%7B5%7D%7D%5Cright%5D%7D" />.