What is the Integral of Cosec x

What is the integral of cosec x?

The cosec or cosecant function is the reciprocal of the sine function and mathematically, we can write it as shown below. 

cosec x = 1/sin x

We can find the integral of cosec x in three ways as shown below.

• Using Partial Fractions

Let us write cosec x = 1/sin x

So, ∫cosec x  dx = ∫(1/sin x)dx

Multiplying with sin x in both numerator and denominator, we write the Right Hand Side as 

Let us assume cos x = u. Differentiating both sides, we get -sin x dx = du. Substituting this in the expression, we get.

Using logarithmic relations, we get the integral as (½) ln|(u-1)/(u+1)| + C. Substituting the value, we get 

• Using Trigonometric Identities

In this method, we use the relation cosec x = 1/sin x ⇒ ∫cosec x dx = ∫(1/sin x)dx

Using the half-angle formula sin 2A = 2 sin A cos A, we write

We could make these changes since tan A = sin A/cos A and sec A = 1/cos A

Let us consider y = tan x/2. Differentiating both sides, we get dy = ½ sec²x/2. Substituting this in the integration expression, we get ∫dy/y = ln|y| + C

Substituting the value of y, we get ∫cosec x dx = ln|tan x/2| + C

• By Substitution Method 

In this method, we multiply and divide cosec x with cosec x – cot x. 

Let us assume z = cosec x – cot x. Then, differentiating both sides, we get dz = -cosec x cot x + cosec²x dx

So, substituting this in the integration expression, we get I = ∫dz/z = ln|z| + C

Now substituting the value of z, we get 

∫cosec x dx = ln|cosec x – cot x| + C

Hence, we have seen how the integral of cosec x can be represented in three ways. The different forms are used depending on the problem at hand.