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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