Integral(dx, 1/x) = 1 + Integral(dx, 1/x)which is true, since the
Integral(...)terms are indefinite integrals. The final step reads:
Subtract the term 'Integral(dx, 1/x)' from both sides to get: 0 = 1The problem lies in cancelling the indefinite integrals. If the integrals were evaluated, they would evaluate to expressions such as f(x) + C (on the left) and f(x) + C' (on the right) for constants of integration C and C'. The steps before the cancellation would read:
Integral(dx, 1/x) = 1 + Integral(dx, 1/x)
f(x) + C = 1 + f(x) + C'
The functions can be cancelled, leaving only an expression involving
the constants of integration. This expression says, basically, that
the evaluated integrals are equal to each other, up to the constants
of integration. Without the constants of integration, the original
"proof" would occur.Moral: don't forget the constants of integration.