Today I read a thread of fake proofs on /r/math. The whole thread is great, but perhaps my favourite is the following “proof” I reproduce here.

We attempt to find:

$$\int \frac{1}{f} \frac{\mathrm{d}f}{\mathrm{d}x}$$

Let $\mathrm{d}u = - \frac{1}{f^2}\mathrm{d}x$ and $v = f$.

Then we use the ordinary method of integration by parts:

$$\int u \mathrm{d}v = uv - \int v \mathrm{d}u$$

Substituting:

$$\int \frac{1}{f} \mathrm{d}f = \frac{1}{f}f - \int f - \frac{1}{f^2} \mathrm{d}f$$

Then simplify:

$$\int \frac{1}{f} \mathrm{d}f = 1 + \int \frac{1}{f} \mathrm{d}f$$

Then we subtract $\int \frac{1}{f} \mathrm{d}f$ from each side and we are left incontrovertibly with:

$$0 = 1 \ \Box$$