import math
import pandas
from pandas import DataFrame
# pandas.set_option('display.height', 5000)
pandas.set_option('display.max_rows', 500)
pandas.set_option('display.max_columns', 500)
pandas.set_option('display.width', 180)

print("Choose function to integrate:")
print(DataFrame.from_records([
    ["cos(x)"],
    ["x ^ 4"],
    ["x ^ 3"],
    ["x ^ 2"],
    ["x"],
    ["sin(x)"]
]))

CHOICE = int(input())

print("You choose to integrate", end=' ')
if CHOICE == 0:
    print("cos(x)")
    f = math.cos
    f1 = math.sin
elif CHOICE == 1:
    print("x ^ 4")
    f = lambda x: x ** 4
    f1 = lambda x: x ** 5 / 5
elif CHOICE == 2:
    print("x ^ 3")
    f = lambda x: x ** 3
    f1 = lambda x: x ** 4 / 4
elif CHOICE == 3:
    print("x ^ 2")
    f = lambda x: x ** 2
    f1 = lambda x: x ** 3 / 3
elif CHOICE == 4:
    print("x")
    f = lambda x: x
    f1 = lambda x: x ** 2 / 2
elif CHOICE == 5:
    print("sin(x)")
    f = lambda x: math.sin(x)
    f1 = lambda x: -math.cos(x)

print ("Enter A and B - boudaries of the segment of integration.")
a, b = (lambda x: (float(x[0]), float(x[1])))(input().split())

print("Enter m - number of division segments.")
m = int(input())

table_xk = [a + (b - a) * i / m  for i in range(0, m + 1) ]

# print("Generated table of x_k with values of f.")
# print(DataFrame.from_records([
#   ["x_k"] + table_xk,
#   ["f(x_k)"] + [f(xi) for xi in table_xk],
# ]))

h = (b - a) / m

def rectangular_formula(f, h, table_xk):
    val = zip(table_xk, table_xk[1:])
    return h * sum([ f((c + d) / 2) for c, d in val ])

def trapez_formula(f, h, table_xk):
    f_val = [f(xk) for xk in table_xk]
    f_val = zip(f_val, f_val[1:])
    return h * sum([ (c + d) / 2 for c, d in f_val ])

def simps_formula(f, h, table_xk):
    x_val = zip(table_xk, table_xk[1:])
    x_val = [ [xk, (xk + xk1) / 2, xk1] for xk, xk1 in x_val ]
    return h / 6 * sum([ f(c) + f(d) * 4 + f(e) for c,d,e in x_val ])

f_int = f1(b) - f1(a)

rec = rectangular_formula(f, h, table_xk)
trap = trapez_formula(f, h, table_xk)
simz = simps_formula(f, h, table_xk)
print('Real integration value is', f_int)
print('Rectangular formula gives value', rec, 'and error', abs(rec - f_int))
print('Trapez formula gives value', trap, 'and error', abs(trap - f_int))
print('Simps formula gives value', simz, 'and error', abs(simz - f_int))