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

The Mathematician — Sun, 22 Jan 2023 13:00:39 +0000

The integral of

\csc^2(ax)

\frac{1}{a}\cot(ax) + C

.

Proof. First we want determine the integral of

\csc^2(x)

, that is:

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

We have seen here that

\frac{d}{dx} \cot(x) = -\csc^2(x)

. Now take the antiderivative of

\csc^2(x)

, which is equal to

-\cot(x)

. Therefore:

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

Lastly, we want to determine the integral of

\csc^2(ax)

. We now that the following derivative is

\frac{d}{dx} \cot(ax) = -a\csc^2(ax)

. So we can take the antiderivative of

\csc^2(ax)

which results:

\begin{align*} \int \csc^2(ax) dx = -\frac{1}{a}\cot(ax) + C. \end{align*}

Therefore, the integral of

\csc^2(ax)

\frac{1}{a}\cot(ax) + C

integral of csc^2(x) Archives - Epsilonify