Deriving Integral of Cosec x

Integral of cosec x is found using various methods that includes:

• Integral of Cosec x By Substitution Method
• Integral of Cosec x By Partial Fraction Method

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]

Integral of the Cosec x is ∫cosec x dx = ln |cosec x – cot x| + C. Cosec is also called the cosecant function. Its integration is denoted as ∫(cosec x).dx. It is one of the fundamental integrations involving trigonometric functions. 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.