Introduction to the integral of csc^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 sine squared. You will also understand how to compute csc^2x integral by using different integration techniques.

What is the integral of csc^2x?

The integration of csc^2x is an antiderivative of csc function which is equal to –cot x + c. It is also known as the reverse derivative of cot function which is a trigonometric identity in calculus. 

The cosecant function is the reciprocal of the tangent function, which is written as:

$\csc x = \frac{1}{\sin x}$ 

Integral of csc2x formula

The formula of integral of csc contains integral sign, coefficient of integration and the function as csc x. It is denoted by ∫(csc2x)dx. In mathematical form, the csc^2x integral is:

$∫\csc^2xdx = - \cot x + c$

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

How to calculate the antiderivative of csc^2x?

The integral of csc^2 is its antiderivative that can be calculated by using different integration techniques. In this article, we will discuss how to calculate integral of cosecant by using:

1. Integration by parts
2. Substitution method

Integral of csc x squared 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 integral of csc squared x by using the integration by parts calculator.

Proof of integral of csc^2x by using integration by parts

To integrate the function csc^2x using integration by parts, we can use the following formula:

$∫udv = uv − ∫ v du$

Let's start by choosing our u and dv terms. In this case, we can choose:

$u = \csc^2x$

$dv = dx$

Next, we need to find the derivatives of u and the antiderivative of dv.

Taking the derivative of u, we get:

$\frac{du}{dx} = -2\cot x \csc^2x$

Taking the antiderivative of dv, we get:

$v = x$

Now we can use the integration by parts formula:

$∫\csc^2x dx = ∫ u dv = uv - ∫ v du$

Substituting in our chosen values for u, v, du/dx, and v, we get:

$∫ csc^2x dx = \csc^2x(x) - ∫ x(-2\cot x \csc^2x) dx$

Simplifying the expression, we get:

$∫\csc^2x dx = x \csc^2x + 2∫\cot x dx$

Since the integral of cot x is, 

$∫\cot x dx = \ln|\sin x|$ 

Substituting it into our original equation, we get:

$∫\csc^2x dx = x \csc^2x + 2 \ln|\sin x| + C$

Therefore, the integral of csc^2 by integration by parts is x csc^2x + 2 ln|sinx| + C.

Integration of csc^2x 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 integral of csc squared by using the substitution method.

Proof of Integral of csc^2x by using substitution method

To proof the antiderivative of csc^2x by using substitution method finder, suppose that:

$I = ∫\csc^2x dx =∫\frac{1}{\sin^2x}dx$

Multiplying and dividing by sin x, 

$I =∫\frac{\sin x}{\sin^3x}dx$

Now since we know that,

$\sin^2x = 1 - \cos^2x$

Using this in the above equation, we get

$I =∫\frac{\sin x}{(1-\cos^2x)^{3/2}} dx$

Now suppose that, 

$u = \cos x$

and

$du = -\sin xdx$ 

then, 

$I =∫\frac{-du}{(1-u^2)^{3/2}}$

Integrating with respect to the variable u, we get,

$I=\frac{-u}{\sqrt{1-u^2}}+c$

Now substituting the value of u, we get

$I=\frac{-\sin x}{\sqrt{1-\sin^2x}}+c$

Since, 

$1-\sin^2x =\cos^2x$

then, 

$I=\frac{-\sin x}{\cos x}+c$

Or, 

$I=-\cot x+c$

Hence the integration of csc^2x is verified by using the substitution method. 

FAQ’s

What is the antiderivative of csc2 X?

The antiderivative or integral of the function csc^2x is equal to -cot x. It can be calculated by using various integration techniques such as, integration by parts formula or the substitution method etc.

What is the formula of cosec?

We know that the cosecant is a trigonometric function which is the reciprocal of other trigonometric function, sine. Therefore, the formula of cosec is written as;

csc x =1/sin x