I can't remember a fallacious proof involving integrals and trigonometric identities.?

Question

My calc professor once taught us a fallacious proof. I'm hoping someone here can help me remember it. Here's what I know about it: - The end result was some variation of 0=1 or 1=2. - It involved (indefinite?) integrals. - It was simple enough for Calc II students to grasp. - The (primary?) fallacy was that the arbitrary constants (+ C) were omitted after integration. I'm not certain, but I have a strong hunch it involved a basic trigonometric identity.

Answer 1

Perhaps it was the integral of the tangent function.

When you integrate tangent by parts you get log secant.

When you integrate sine over cosine by parts you get log cosine.

Using limits that cross pi/2 causes one to blow up but not the other

Answer 2

You can do this with integration by parts for ∫1/x dx:

u = 1/x

dv = dx

then

du = -1/x² dx

v= x

∫1/x dx = (1/x)(x) - ∫(x)(-1/x²) dx

∫1/x dx = 1 + ∫1/x dx

then cancel the indefinite integrals on both sides:

0 = 1.

The correct step to take would be to express the indefinite integrals as the antiderivative function plus a constant of integration:

∫1/x dx = 1 + ∫1/x dx

F(x) +C₁ = 1+ F(x)+C₂

C₁ = 1 + C₂

(the constants of integration are not required to be equal on both sides)