Integral of Csc x

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

Alan Walker-

Published on 2023-04-14

Introduction to integral of csc

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 cosecant. You will also understand how to compute cosecant integral by using different integration techniques.

What is the integral of cscx?

The csc integral is an antiderivative of the cosine function which is equal to ln |cosec x – cot x|. It is also known as the reverse derivative of the cosine function which is a trigonometric identity. The cosine function is the ratio of the opposite side to the hypotenuse of a triangle which is written as:

cos = adjacent side/hypotenuse

The integral of csc(x) is a common integral in calculus. It involves a trigonometric function cosecant(x). It is used to solve many integral problems such as to solve the integral of csc square x.

Integration of csc formula

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

$\int \csc xdx = \ln |\csc x – \cot x| + c{2}lt;/p>

Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of the integral. Using the above formula, we can evaluate the integral of tan x.

How to calculate the integral of cosecant(x)?

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

  1. Substitution method
  2. Partial Fraction

Integral of csc x by using substitution method

The substitution method involves many trigonometric formulas. We can use these formulas to verify the integrals of different trigonometric functions such as sine, cosine, tangent, etc. Let’s understand how to prove the csc x integral by using the substitution method.

Proof of integral of cscx by using substitution method

To integrate csc x by using the u-substitution method calculator, suppose that:

$I = \int \csc xdx{2}lt;/p>

Multiplying and dividing by csc x – cot x, we get

$I=\int \frac{\csc x(\csc x – \cot x)}{(\csc x – \cot x)}dx{2}lt;/p>

Solving further,

$I = ∫{(\csc^2 x – \csc x.\cot x)/(\csc x – \cot x)}dx{2}lt;/p>

Or, it can be written as:

$I=\int \frac{(– \csc x.\cot x + \csc^2x )}{(\csc x – \cot x)}dx{2}lt;/p>

Here f(x) = csc x – cot x and it’s derivative as f’(x) = – csc x.cot x + csc2x, therefore, the above integral can be written as;

$I = \int \frac{f’(x)}{f(x)}dx{2}lt;/p>


$I = \ln |f(x)| + c{2}lt;/p>

Using the value of f(x),

$I = \ln |\csc x – \cot x| + c{2}lt;/p>

Hence we have verified the integral of cosecant x. Some integrals can also be evaluated by using the trigonometric substitution calculator.

Integration of csc by using partial fraction

A partial fraction is used to decompose rational expressions. It is also used to find the integral of any rational function easily. Therefore we can use this method to calculate the integration of cscx.

Proof of csc integral by using partial fraction

To prove the csc x integral,

$\int \csc xdx = \int \frac{1}{sin x}dx{2}lt;/p>

Multiplying and dividing this by sin x,

$\int \csc xdx = \int \frac{\sin x}{\sin^2x}dx{2}lt;/p>

Using one of the trigonometric formulas,

$\int \csc xdx =\int \frac{\sin x}{(1 - \cos^2x)}dx{2}lt;/p>

Now, assume that cos x = u. Then -sin x dx = du. Then the above integral becomes,

$\int \csc xdx=\int \frac{du}{(u^2 - 1)}{2}lt;/p>

Now by partial fractions,

$\frac{1}{(u^2 - 1)}= \frac{1}{2}\left[\frac{1}{(u - 1)} - \frac{1}{(u + 1)}\right].{2}lt;/p>


$\int \csc xdx=\frac{1}{2}\int \left[\frac{1}{(u - 1)} - \frac{1}{(u + 1)}\right]du{2}lt;/p>

$\int \csc xdx = \frac{1}{2}\left[\ln|u - 1| - \ln|u + 1|\right] + C{2}lt;/p>

By using logarithmic laws,

$\int \csc xdx = \frac{1}{2}\left[\ln \left|\frac{(u – 1)}{(u + 1)}\right|\right] + C{2}lt;/p>

Now substituting the value of u,

$ \int \csc xdx= \frac{1}{2}\left[\ln \left|\frac{(\cos x – 1)}{(\cos x + 1)}\right|\right] + C{2}lt;/p>

Hence we have verified the integral of cosecant by using the integration by partial fraction.

Integral of csc x by using integration by parts

The derivative of a function calculates the rate of change, and integration is the process of finding the antiderivative of a function. The integration by parts is a method of solving the integral of two functions combined together. Let’s discuss calculating the integration of csc x by using the integration by parts.

Proof of integral of csc x by using integration by parts

Since we know that the function cosec x can be written as the product of two functions. Therefore, we can calculate the integral of cscx by using integration by parts. For this, suppose that:

$I =\int \csc xdx=\int 1.\csc x dx{2}lt;/p>

By using the integration by parts rule calculator, we know that,

$\int [f(x).g(x)]dx = f(x).\int g(x)dx - \int [f’(x).\int g(x)]dx{2}lt;/p>


$\int \csc x dx = x.\csc x – \int(-x\cot x.\csc x)dx{2}lt;/p>

$\int \csc x dx= x.\csc x + [x.\csc x - \int \csc x dx]{2}lt;/p>

Since I = ∫ cosec x dx, so,

$2I = 2x.\csc x{2}lt;/p>


$I = x.\csc x{2}lt;/p>

This method is also applicable to calculate the integral of sec x.


What is the integral of csc(x)?

The csc x integral is the natural logarithm of the absolute value of the cosecant of x, plus a constant of integration. In other words, ∫csc(x) dx = ln|csc(x)| + C, where C is an arbitrary constant.

How do you evaluate the integral of csc(x)?

There are several methods for evaluating the integral of csc(x), including substitution, integration by parts, and trigonometric identities. One common technique is to use the substitution u = sin(x), which transforms the integral into ∫csc(x) dx = ∫(1/sin(x)) dx = ∫(1/u)(du/sqrt(1 - u^2)) = ln|csc(x) + cot(x)| + C.


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