What is the integral of cot^2(x)?

integral of cot^2(x)

The integral of

\cot^2(x)

- x - \cot(x)

.

Proof. We want to determine the integral of

\cot^2(x)

\begin{align*} \int \cot^2(x) dx. \end{align*}

We have seen here that

\cot^2(x) = \csc^2(x) - 1

. So we get:

\begin{align*} \int \cot^2(x) dx &= \int (\csc^2(x) - 1)dx \\ &= \int \csc^2(x)dx - \int 1\cdot dx. \end{align*}

Further, we saw here that the integral of

\csc^2(x)

-\cot(x)

plus some constant. So, we get:

\begin{align*} \int \cot^2(x) dx &= \int (\csc^2(x) - 1)dx \\ &= \int \csc^2(x)dx - \int 1\cdot dx \\ &= -\cot(x) - x + C. \end{align*}

Therefore, the integral of

\cot^2(x)

-\cot(x) - x