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

integral of tan^2(x)

The integral of

\tan^2(x)

\tan(x) - x + C

.

Proof. We want to determine the integral of

\tan^2(x)

, that is:

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

We have seen here the next trigonometric identity:

\begin{align*} \tan^2(x) + 1 = \sec^2(x). \end{align*}

So we get the following integral:

\begin{align*} \int \tan^2(x) dx = \int (\sec^2(x) - 1)dx. \end{align*}

Further, we have seen that the integral of

\sec^2(x)

\tan(x)

plus some constant. Therefore, we get the following:

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

So, the integral of

\tan^2(x)

\tan(x) - x + C