Integral of Cot X

Integral of cotx along with its formula and proof with examples. Also learn how to calculate integration of cot x with step by step examples.

Alan Walker-

Published on 2023-04-20

Introduction to the integral of cotx

In calculus, the integral is a fundamental concept that assigns numbers to functions to define displacement, area, volume, and all those functions that contain a combination of tiny elements. It is categorized into two parts, definite integral and indefinite integral calculator. The process of integration calculates the integrals. This process is defined as finding an antiderivative of a function. 

Integrals can handle almost all functions, such as trigonometric, algebraic, exponential, logarithmic, etc. This article will teach you what is integral to a trigonometric function cot. You will also understand how to compute integral of tan by using different integration techniques.

What is the integral of cot?

The integral of cot x is an antiderivative of cotangent function which is equal to ln|sin x|. It is also known as the reverse derivative of sine function which is a trigonometric identity. 

The cotangent function is the reciprocal of tangent function, that is;

$\cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x}$ 

Integral of cotx formula

The formula of integral of cotangent contains integral sign, coefficient of integration and the function as cot. It is denoted by ∫(cot x)dx. In mathematical form, the integration of cot x is:

$∫\cot xdx = \ln|\sin x| + c$

Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of integral. 

How to calculate the integration of cotx?

The integral of cotangent is its antiderivative that can be calculated by using different integration techniques. In this article, we will discuss how to calculate integral of tangent by using:

  1. Substitution method
  2. Definite integral

Integration of cot by using u substitution

The u substitution is a method of integration in calculus. It is used to calculate integral of a function which is complex to be calculated by usual integration. Let’s discuss calculating the antiderivative of cotx by using u-substitution method calculator.

Proof of integral of cot x by using u-substitution

Since we know that a function can be replace by u in u-substitution. Therefore, we can calculate the integral of cot by using this technique. 

Assume that,

$I = ∫\cot xdx$

By using trigonometric identities, cot x can be written as cos x/sin x. Then,

$I = ∫\frac{\cos x}{\sin x}dx$

Now, suppose that u = sin x and du = cos x dx,


$I = ∫\frac{1}{u}du$ 


$I = \ln|u|+c$

Substituting the value of u, we get 

$I = \ln|\sin x| + c$

Which is the calculation of the integral of cot x.

Antiderivative of cotx by using definite integral 

The definite integral is a type of integral that calculates the area of a curve by using infinitesimal area elements between two points. The definite integral can be written as:

$∫^b_a f(x) dx = F(b) – F(a)$

Let’s understand the verification of the integration of cotx by using the definite integral.

Proof of integral of cotangent by using definite integral

To compute the integral cot x by using a definite integral calculator, we can use the interval from 0 to π/2 and π/4 to π/2. Let’s compute the integration of cot x from 0 to π/2. For this we can write the integral as:

$∫^{\frac{π}{2}}_0 \cot x dx = \ln|\sin x|^{\frac{π}{2}}_0$ 

Now, substituting the limit in the given function.

$∫^{\frac{π}{2}}_0 \cot x dx = \ln \sin (π/2) – \ln \sin (0)$

Since sin 0 is equal to 0 and sin π/2 is equal to 1, therefore, 

$∫^{\frac{π}{2}}_0 \cot x dx = \ln(1) –\ln(0)= 0$

Which is the calculation of the definite integral of cot. Now to integrate cot x between the interval π/4 to π/2, 

$∫^{\frac{\pi}{2}}_{\frac{\pi}{4}} \cot x dx = \ln(\sec x)|^{\frac{\pi}{2}}_{\frac{\pi}{4}}$


$∫^{\frac{\pi}{2}}_{\frac{\pi}{4}}\cot x dx = \ln(\sin\frac{\pi}{2}) –\ln(\sin\frac{\pi}{2})$

Since sin π/2 is equal to 1 and sin π/4 is equal to 1/√2, therefore, 

$∫^{\frac{\pi}{2}}_{\frac{\pi}{4}}\cot x dx = \ln(1)- \ln(1/√2)= ln 1 - ln 1 + ln √2$

$∫^{\frac{\pi}{2}}_{\frac{\pi}{4}}\cot x dx =ln √2$

Therefore, the definite integral of cotx is equal to ln √2. Try our double integral calculator to integrate cot x twice.


What is the antiderivative of cot X?

The antiderivative or the cot x integration is equal to ln|sin x|. It can be calculated by using different integration techniques such as u-substitution or definite integral. Mathematically, the integral of cot x is written as;

$\int \cot x=\ln |\sin x|+c$

What is cot x equal to?

The cot x is a trigonometric function which is equal to the ratio between cos x and sin x. Mathematically, it can be expressed as;

$\cot x=\frac{\cos x}{\sin x}$

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