Question

How do you find the integral of \[\int{\cot xdx}\] ?

Answer 1

Hint:In the given question, we have been asked to integrate the given constant. In order to solve the question, we integrate the given expression by using the basic concept of integration. First we need to integrate the given expression. Later we will need to integrate by substituting t = sin(x) then differentiate with respect to ‘x’ using a suitable integration formula and simplify further. In this way we will get our required answer.

Complete step by step answer:
We have given,
\[\Rightarrow \int{\cot xdx}\]
Let I be the integration of the given equation.
Therefore, we can write the integration as,
\[\Rightarrow I=\int{\cot xdx}\]
As we know that,
Integration of cot(x) by using the integration formulas of trigonometric,
\[\Rightarrow I=\int{\cot xdx}=\int{\dfrac{\cos x}{\sin x}dx}\]
Now,
Substitute \[t=\sin x\Rightarrow \dfrac{dt}{dx}=\cos x\Rightarrow dt=\cos xdx\]
Therefore,
\[\Rightarrow I=\int{\dfrac{dt}{t}}\]
Using the standard integral formula, we get
\[\Rightarrow I=\ln \left( t \right)+C\]
Undo the substituting in the above expression, we get
\[\Rightarrow I=\ln \left| \sin x \right|+C\]
\[\therefore \int{\cot xdx}=\ln \left| \sin x \right|+C\]

Hence,the integral of \[\int{\cot xdx}\] is $\ln \left| \sin x \right|+C$.

Note:This is the easiest way to solve this question, you have to convert the equation in the terms of sin(x) and cos(x) as it will become easy to integrate the given trigonometric function. Integrals of trigonometric functions are already defined but of basic identity, if you find any changes in the angle given then you have to first expand or convert the given integral in the basic simplest form if trigonometry and then further integrate the given integral. Always keep in mind some standard integral as it will directly give you an answer, you don’t need to integrate the given expression further.