Proof of 0 = 1

ʃ tan(x) dx = ʃ sin(x) / cos(x) dx

ʃ tan(x) dx = ʃ sec(x) sin(x) dx [taking sec(x) as the 1st function]
ʃ tan(x) dx = - sec(x) cos(x) + ʃ sec(x) tan(x) cos(x) dx
ʃ tan(x) dx = - 1 + ʃ tan(x) dx
0 = - 1
0 + 1 = - 1 + 1
1 = 0