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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

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

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

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

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:

$$\int \csc^{2}x \ dx – 1 \int \ dx$$

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:

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

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

$$-\cot x – x$$

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