Integral of the Cosecant function, denoted as ∫(cosec x).dx is the fundamental integration in trigonometry. Cosec x is the reciprocal function of sin x. In this article, we will understand the formula of the integral of cosec x and the Methods of Integral of cosec x.
Integral of Cosec x Definition
Integral of cosec x with respect to (x) is denoted as: ∫csc(x).dx. And its value is given by:
∫csc(x).dx = ln ∣csc(x) – cot(x)∣ + C
where,
- C represents Constant of Integration
- ln denotes Natural Logarithm
Integral is also known as antiderivative of cosec x.
We have multiple formulas for this integral, but one of the more popular ones is:
∫csc(x).dx = ln ∣csc(x) – cot(x)∣ + C
Additionally, we can express the integral using other equivalent forms:
([Tex]\int \csc(x) . dx = -\ln | \csc(x) + \cot(x) | + C[/Tex])
[Tex](\int \csc(x) . dx = \frac{1}{2} \ln \left| \frac{\cos(x) – 1}{\cos(x) + 1} \right| + C)[/Tex]
[Tex](\int \csc(x) . dx = \ln \left| \tan\left(\frac{x}{2}\right) \right| + C)[/Tex]
Cosec x Function
Cosec x is the reciprocal of the sine function. It is defined as, csc(x) = 1/sin(x)
Integral of the cosecant function, denoted as ∫cosec(x)dx, signifies the accumulated area beneath the cosecant curve from a designated starting point to a particular endpoint along the x-axis. Mathematically, it is typically represented as:
∫cosec(x)dx = ln |cosec(x) – cot(x)| + C
Cosec x integral applications extend across physics, engineering, and mathematics, playing a crucial role in the analysis of periodic functions and waveforms.
Let’s have a look at the formula for the integral of cosec x with respect to x is shown below.
∫ cosec x dx = ln |cosec(x) – cot(x)| + C
Integral of cosec is found using various methods and some important of them are added below in the article.
Deriving Integral of Cosec x
Integral of cosec x is found using various methods that includes:
Integral of Cosec x By Substitution Method
To find the integral of cosec x using the substitution method, we multiply and divide the integrand by (csc(x) – cot(x)). Let’s see how this works step by step:
[Tex][ \int \csc(x) . dx ~=~ \int \frac{\csc(x) \cdot (\csc(x) – \cot(x))}{\csc(x) – \cot(x)} . dx ][/Tex]
Assume {csc(x) – cot(x) =u}. Then (-csc(x)cot(x) + csc 2 (x).dx = du)
Substituting these values,
[Tex][ \int \csc(x) . dx~=~\int \frac{du}{u} = \ln |u| + C[/Tex]
Substituting back {u = csc(x) – cot(x)},
∫csc(x).dx = ln ∣csc(x) – cot(x)∣ + C
Integral of Cosec x By Partial Fraction Method
Integral of cosec x is also found using partial fractions. Since csc(x) is the reciprocal of sin(x), we have:
csc(x) = 1/sin(x)
Start with:
[Tex][ \int \csc(x).dx ~= ~\int \frac{1}{\sin(x)}. dx [/Tex]
Multiply and divide by {sin(x)}
[Tex][ \int \csc(x) . dx ~=~ \int \frac{\sin(x)}{\sin^2(x)} . dx ][/Tex]
Using trigonometric identity sin2(x) = 1 – cos2(x), we get,
[Tex]\int \csc(x) . dx ~=~ \int \frac{\sin(x) . dx}{1 – \cos^2(x)}[/Tex]
Assume {cos(x) = u}, then {-sin(x).dx = du}
Substituting these values,
[ [Tex]\int \csc(x).dx~=~\int \frac{du}{1 – u^2} [/Tex]]
Integral of ([Tex]\frac{1}{1 – u^2}[/Tex]) can be expressed as a natural logarithm:
[[Tex] \int \frac{du}{1 – u^2}~=~\frac{1}{2} \ln \left| \frac{1 + u}{1 – u} \right| + C[/Tex] ]
Substituting back (u = cos(x)), we obtain:
[Tex]\int \csc(x).dx~=~\frac{1}{2} \ln \left| \frac{1 + \cos(x)}{1 – \cos(x)} \right| + C[/Tex]
Applications of Integral of Cosec x
The integral of cosec x finds applications in various fields. Here are some notable ones:
- Physics and Engineering: In wave theory, the csc x integral is used to study different types of waves such as sound waves and electromagnetic waves. It helps to determine the vibration frequency, frequency and phase.
- Electrical Engineering: the core of csc x lies in the analysis of alternating current (AC) circuits. It helps in calculating/finding power factor, impedance and phase angle.
- Mathematics: Trigonometric integrals such as the integral of csc x are important in solving differential equations, evaluating complex fractions, and understanding periodic functions
Definite Integral of Cosec x
By the fundamentals properties of definite integrals, we can calculate the definite integral of cosec x within any two time intervals limits.
Definite Integral of f(x) = [Tex]\int_{a}^{b} f(x).dx~=~F(b)~-~F(a)[/Tex]
Definite Integral of cosec within ‘a’ to ‘b’ interval = [Tex]\int_{a}^{b} \csc(x).dx~=~ln|cosec(x)~-~cot(x)|_{b}^{a}~=~[ln|cosec(b)~-~cot(b)|~-~ln|cosec(a)~-~cot(b)|][/Tex]
Also, Check
Examples on Integral of Cosec x
Example 1: Evaluate ∫csc(x).dx
Solution:
= ∫csc(x).dx
= ln ∣csc(x) – cot(x)∣ + C
Example 2: Find the integral [Tex] \int \frac{\csc x}{\sin x}.dx[/Tex]
Solution:
[Tex] \int \frac{\csc x}{\sin x}.dx[/Tex]
We can rewrite the integrand as:
[Tex]\frac{\csc x}{\sin x}~=~\frac{1}{\sin^2 x}[/Tex]
Now,
[Tex]\int \frac{1}{\sin^2 x}.dx~=~-\cot x + C[/Tex]
Example 3: Evaluate the integral: [Tex]\int \frac{\csc x}{\cot x}.dx[/Tex]
Solution:
= [Tex]\int \frac{\csc x}{\cot x}.dx[/Tex]
We can rewrite the integrand as:
[Tex][ \frac{\csc x}{\cot x} = \frac{\sin x}{\cos x} = \tan x ][/Tex]
Now,
[Tex]\int \tan x , dx~=~-\ln |\cos x| + C[/Tex]
Example 4: Calculate the integral: [Tex]\int \frac{\csc^2 x}{\cot x}.dx[/Tex]
Solution:
= [Tex]\int \frac{\csc^2 x}{\cot x}.dx[/Tex]
We know that [Tex](\csc^2 x = 1 + \cot^2 x)[/Tex].
Therefore:
[Tex]\int \frac{\csc^2 x}{\cot x}.dx\\=~\int \frac{1 + \cot^2 x}{\cot x}.dx \\=~\int (\cot x + \cot^3 x).dx[/Tex]
Integrating each term separately:
[Tex]\ln |\sin x|~-~\frac{1}{2} \cot^2 x~+~C[/Tex]
Practice Problems on Integral of Cosec x
P1: Evaluate ∫csc2(x).dx
P2: Find [Tex]\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \csc(x).dx[/Tex].
P3: Evaluate the following integral: ∫csc(x)cot(x).dx
P4: Find the indefinite integral of csc(x) using the partial fractions method.
FAQs on Integral of Cosec x
What is the integral of cosec x?
Integral of cosec x is given by[Tex]\int \csc(x).dx~=~\ln |\csc(x)~-~\cot(x)|+C[/Tex]
How to find antiderivative of Cosec x?
Antiderivative of cosec x, ∫ cosec x dx and its value is, ∫cosec x dx = ln |cosec x – cot x| + C
How do you solve trigonometric integrals?
Solution to trigonometric integrals can be done using various techniques including substitution, partial fractions, and trigonometric identities.
Where is integral of cosec x used in real life?
Integral of cosec x is used in applied in physics, engineering and mathematics to analyze waveforms, AC circuits and periodic functions.
How do you graph function (csc x)?
To graph (csc x), plot points where the function is undefined (vertical asymptotes) at multiples of πand π/2, then sketch the curve following the behavior around these asymptotes.
What is the period of the function csc(x)?
Period of (cscx) is 2π.
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