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Integral of cotangent

The formula of integral of cot is:

\displaystyle \int \cot u \cdot du = \ln |\sin u|+ \text{C}

Let’s see some examples for integrals of cotangent.

Example 1. Integral of cot 2x

\displaystyle \int \cot(2x) \ dx=

We substitute the 2x for u, we derive and we pass dividing the 2:

u = 2x \quad \Rightarrow \quad du = 2 \ dx \quad \Rightarrow \quad \cfrac{du}{2} = dx

And we replaced the terms for u:

\displaystyle \int \cot(u) \cfrac{du}{2}

By properties of integrals we extract the \frac{1}{2} from the integral:

\displaystyle \cfrac{1}{2}\int\cot(u) \ du

Now we directly apply the formula of the integral of cotangent:

\displaystyle \cfrac{1}{2}\int \cot (u) \ du = \left (\cfrac{1}{2}\right)(\ln \sin(u))

And finally we substitute the u for 2x and the answer will be:

\cfrac{1}{2} \ln \left| \sin(2x)\right|

Example 2. Integral of square cotangent

\displaystyle \int \cot^{2}(x) \ dx

The fastest way to do this integral is to review the formula in the integrals form and you’re done. Another way is to shred the integral a little and review the integral form anyway at some point, let’s start:

To begin with the resolution of this integral, the first thing we have to do is apply the following trigonometric identity:

\cot^{2}x + 1 = \csc^{2}x

Now what you have to do is isolate \cot^{2}x:

\cot^{2}x = \csc^{2}x  - 1

Substituting \cot^{2}x in the integral, we will obtain the integral of a subtraction:

\int\left ( \csc^{2}x  - 1 \right) \ dx

Separate the integral to a sum of integrals:

\displaystyle \int \csc^{2}x \ dx + \int - 1 \ dx

Applying properties of the integrals we will remove the -1 from the integral:

\displaystyle \int \csc^{2}x \ dx - 1 \int \ dx

To solve the first integral, we will review the integral form that shows us an integral formula of \csc^{2}x, therefore, the first integral would be as follows:

\displaystyle \int\csc^{2}x \ dx- \int \ dx = -\cot x - \int \ dx

Solving the second integral, the answer will be the following:

-\cot x - x

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