Integral of Cot^2x

Integral of cot^2x along with its formula and proof with examples. Also learn how to calculate integration of cot^2x with step by step examples.

Alan Walker-

Published on 2023-05-03

Introduction to the integral of cot^2x

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. 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 the integration of cot^2x by using different integration techniques.

What is the integration of cot square x?

The integral of cot^2x is an antiderivative of cotangent squared function which is equal to -cot x - x +c. It is also known as the reverse derivative of cot square which is a trigonometric identity. The calculation of the cot square integration can be easy and simple by using our integral calculator.

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

cot x = 1/ tan x = cos x/ sin x 

Integral cot square x formula

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

 ∫(cot^2 x)dx = -cot x - x + c

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

How to calculate the integral of cot^2(x)?

The integral of cot x 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. Trigonometric formulas
  2. Definite integral

Integral of cot x square by using trigonometric formulas

This method is used to evaluate integrals by using different trigonometric formulas in calculus. In this method, we can replace the original function with any other trigonometric formula. Let’s discuss calculating the integral of cot x squared by using trigonometric substitution calculator.

Proof of integral of cot^2x by using trigonometric formulas

Since we know that a function can be replaced by another function. Therefore, we can calculate the integration of cot square x by using this technique. 

Assume that,

I = ∫(cot^2x)dx

By using trigonometric formula,


By rearranging, 

cot^2x = csc^2x - 1

Now use this formula in the above integral, 

I = ∫(csc^2x - 1)dx

Integrating each part of the integral, we get

I = -cot x-x+c

Which is the calculation of the integral of cot square x. Use our indefinite integral calculator to make your calculations easy and quick.

Integration of cot^2x 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:

∫abf(x) dx = F(b) – F(a)

Let’s understand the verification of the integration of cot square x by using the definite integral.

Proof of integral of cot^2x by using definite integral

To compute the integral cot^2(x) by using a definite integral calculator, we can use the interval from 0 to π/3. Let’s compute the integral of cot square x from 0 to π/3. For this we can write the integral as:

∫0π/3cot^2 x dx = -cot x-x|0π/3 

Now, substituting the limit in the given function.

∫0π/3cot x dx = -cot π/3 -π/3 – (-cot 0 - 0)

Since cot 0 is equal to 0 and cot π/3 is equal to 1/√3, therefore, 

∫0π/3cot x dx = -1/√3 –π/3

Which is the calculation of the definite integral of cot square x.


How do you integrate cot squared?

To integrate cot squared, the trigonometric formulas can be used. For this, replace the function cot^2x with the trigonometric formula, cot^2x=csc^2x-1 and then integrate each term. Mathematically, 

$\int \cot^2x=-\cot x-x+c$

What is cos2x integral?

The integral of cos 2x can be calculated by using usual integration techniques and integration formulas. It can be written as;

$\int \cos 2x=\frac{\sin 2x}{2}+c$

Related Problems

Copyright © 2023