# IBP Paradox for cot x integral

Greetings! I hope this is the correct forum for my question (this is my first post, here).

The problem statement

Find: $\displaystyle\int \cot x \ \mathrm{d}x$

A Solution
We can re-write this integral into a convenient form: $\displaystyle\int \cot x \ \mathrm{d}x = \displaystyle\int \dfrac{\cos x}{\sin x} \ \mathrm{d}x$

Now, let $u=\sin x$. Then, $du = \cos x \ \mathrm{d}x$.

The integral becomes: $\displaystyle\int \dfrac {du}{u} \ \mathrm{d}x$ and we integrate to get $\ln |u|$ (ignoring the constant of integration, for now -- reason: simplicity).

We convert our result into a formula with respect to $x$:
$\boxed{\ln |\sin x| + \mathcal{C}}$​

My Question

If we use Integration by Parts, however, with $u=\dfrac{1}{\sin x}$ and $\mathrm{d}v = \cos x \ \mathrm{d}x$ and
$\displaystyle\int u \ \mathrm{d}v = uv - \displaystyle\int v \ \mathrm{d}u,$​
we get $1=0$. I can show my solution, if needed. Why is this? (A friend of mine pointed this out to me)

Related Calculus and Beyond Homework Help News on Phys.org
dynamicsolo
Homework Helper
You get 1 = 0 only by rather slippery neglect of the arbitrary constant. The full integration-by-parts would have to read

$$\int \frac{cos x dx}{sin x} = ( csc x )( sin x ) + \int (sin x) ( csc x cot x ) dx \mathbf{+ C } .$$

Someone had a post a month or so back (I don't know if I can find it again quickly) where the paradox there also turned on making a "1" appear out of nowhere. What you learn about indefinite integration is that finite numbers don't count for much...

That sounds facetious, but you do find that, for example, certain trigonometric-powers integrals can give one form of an anti-derivative using one technique of integration, and a different (but equivalent) form of the anti-derivative using another method. When you apply trig identities to show the equivalence of the two results, you find that they differ by a small number. But since both results are general anti-derivatives with arbitrary constants, the small number is actually irrelevant (in a sense, it shows the difference in "vertical shift" of the two anti-derivative functions). So the specified constant value can be made to appear or disappear, depending on the choice of anti-derivative. One could say that arbitrary constants of integration "swallow" numerical constants.

Last edited: