from a handpicked tutor in LIVE 1to1 classes
Integral of Cosec x
To find the integral of cosec x, we will have to use trigonometry. Cosec x is the reciprocal of sin x and cot x is equal to (sin x)/(cos x). We can do the integration of cosecant x in multiple methods such as:
 By substitution method
 By partial fractions
 By trigonometric formulas
We have different formulas for integration of cosec x and let us derive each of them using each of the abovementioned methods. Also, we will solve some examples related to the integral of cosec x.
What is the Integral of Cosec x?
The integral of cosec x is denoted by ∫ cosec x dx (or) ∫ csc x dx and its value is ln cosec x  cot x + C. This is also known as the antiderivative of cosec x. We have multiple formulas for this. But the more popular formula is, ∫ cosec x dx = ln cosec x  cot x + C. Here "ln" represents the natural logarithm and 'C' is the constant of integration. Multiple formulas for the integral of cosec x are listed below:
 ∫ cosec x dx = ln cosec x  cot x + C [OR]
 ∫ cosec x dx =  ln cosec x + cot x + C [OR]
 ∫ cosec x dx = (1/2) ln  (cos x  1) / (cos x + 1)  + C [OR]
 ∫ cosec x dx = ln  tan (x/2)  + C
We use one of these formulas according to necessity.
We will prove each of these formulas in each of the mentioned methods.
Integral of Cosec x by Substitution Method
We can find the integral of cosec x proof by substitution method. For this, we multiply and divide the integrand with (cosec x  cot x). But why do we need to do this? Let us see.
∫ cosec x dx = ∫ cosec x · (cosec x  cot x) / (cosec x  cot x) dx
= ∫ (cosec^{2}x  cosec x cot x) / (cosec x  cot x) dx
Now assume that cosec x  cot x = u.
Then (cosec x cot x + cosec^{2}x) dx = du.
Substituting these values in the above integral,
∫ cosec x dx = ∫ du / u = ln u + C
Substituting u = cosec x  cot x back here,
∫ cosec x dx = ln cosec x  cot x + C
Hence we proved the first formula of integral of csc x.
Let us derive another formula from this formula by using the fact cosec x  cot x = 1 / (cosec x + cot x). Then
∫ cosec x dx = ln 1/(cosec x + cot x) + C
= ln (cosec x + cot x)^{1} + C
By a property of logarithms, ln a^{m} = m ln a. So
∫ cosec x dx =  ln cosec x + cot x + C
Hence we have proved the second formula of integral of csc x.
Integral of Cosec x by Partial Fractions
To find the integration of cosec x proof by partial fractions, we have to use the fact that cosec x is the reciprocal of sin x. i.e., cosec x = 1/(sin x). Wait! How can this be turned into partial fractions? Let us see.
∫ cosec x dx = ∫ 1/(sin x) dx
Multiplying and dividing this by sin x,
∫ cosec x dx = ∫ (sin x) / (sin^{2}x) dx
Using one of the trigonometric identities,
∫ cosec x dx = ∫ (sin x dx) / (1  cos^{2}x) dx
Now, assume that cos x = u. Then sin x dx = du. Then the above integral becomes
∫ cosec x dx = ∫ du / (u^{2}  1)
By partial fractions, 1/(u^{2}  1) = 1/2 [1/(u  1)  1/(u + 1)]. Then
∫ cosec x dx = (1/2) ∫ [ 1/(u  1)  1/(u + 1) ] du
= (1/2) [ ln u  1  ln u + 1 ] + C
(We can get this by finding the integrals ∫ [ 1/(u  1) ] du and ∫ [ 1/(u + 1) ] du separately by substituting 1 + u = p and 1  u = q respectively).
By a property of logarithms, ln m  ln n = ln (m/n). So
∫ cosec x dx = (1/2) ln  (u  1) / (u + 1)  + C
Substituting u = sin x back here,
∫ cosec x dx = (1/2) ln  (cos x  1) / (cos x + 1)  + C
Hence proved.
Integral of Cosec x by Trigonometric Formulas
We can prove that the integral of cosec x to be ln  tan (x/2)  + C by using trigonometric formulas. We can write cosec x as 1/(sin x).
\(\int \operatorname{cosec} x d x=\int \frac{1}{\sin x} d x\)
Using halfangle formulas, sin A = 2 sin A/2 cos A/2. Applying this,
\(\int \operatorname{cosec} x d x=\int \frac{1}{2 \sin \frac{x}{2} \cos \frac{x}{2}} d x\)
Multiply and dividing the denominator by cos x/2,
\(\begin{aligned}
\int \operatorname{cosec} x d x &=\int \frac{1}{2\left(\frac{\sin \frac{x}{2}}{\cos \frac{x}{2}}\right) \cos ^{2} \frac{x}{2}} d x \\
&=\int \frac{\left(\frac{1}{2}\right) \sec ^{2} \frac{x}{2}}{\tan \frac{x}{2}} d x
\end{aligned}\)
Now, assume that tan (x/2) = u. From this, (1/2) sec^{2}(x/2) dx = du.
∫ cosec x dx = ∫ du/u = ln u + C
Substituting u = tan (x/2) back,
∫ cosec x dx = ln  tan (x/2)  + C
Hence proved the formula of integral of csc x.
Important Notes Related to Integration of Csc x:
Here are the formulas of integral of csc x with the respective methods of proving them.
 Using the substitution method,
∫ csc x dx = ln csc x  cot x + C (or)
∫ csc x dx =  ln csc x + cot x + C  Using partial fractions,
∫ csc x dx = (1/2) ln  (cos x  1) / (cos x + 1)  + C  Using trigonometric formulas,
∫ csc x dx = ln  tan (x/2)  + C
Topics Related to Integral of Cosec x:
Here are some topics that you may find helpful while doing the integration of cosec x.
Solved Examples on Integration of Cosec x

Example 1: Evaluate the definite integral ∫₀^{π/2 }cosec x dx if it converges.
Solution:
The integral of cosec x is,
∫ cosec x dx = ∫ cosec x dx = ln cosec x  cot x + C
∫₀^{π/2 }cosec x dx is obtained by applying the limits 0 and π/2 for this. Then
∫₀^{π/2 }cosec x dx = [ ln cosec π/2  cot π/2 + C ]  [ ln cosec 0  cot 0 + C ] = Diverges
This is because cosec 0 = ∞.
Answer: ∫₀^{π/2 }cosec x dx diverges.

Example 2: What is the value of ∫ (csc x  cot x) dx?
Solution:
We know that the integrals of cosecant x and cot x are ln csc x  cot x and ln sin x respectively. Thus,
∫ (csc x  cot x) dx = ∫ csc x dx  ∫ cot x dx
= ln csc x  cot x  ln sin x + C
Answer: ∫ (csc x  cot x) dx = ln csc x  cot x  ln sin x + C.

Example 3: Find the value of ∫ (cosec^{2}x + cosec x) dx.
Solution:
∫ (cosec^{2}x + cosec x) dx = ∫ cosec^{2}x dx + ∫ cosec x dx
We know that ∫ cosec^{2}x dx = cot x and ∫ cosec x dx = ln cosec x  cot x + C. So
∫ (cosec^{2}x + cosec x) dx = cot x + ln cosec x  cot x + C
Answer: ∫ (cosec^{2}x + cosec x) dx = cot x + ln cosec x  cot x + C.
FAQs on Integral of Cosec x
What is the Integral of Cosec x?
There are multiple formulas for the integral of cosec x. But the most popular formula that we use is ∫ cosec x dx = ln cosec x  cot x + C, where C is the integration constant.
How do You Find the Antiderivative of Cosec x?
The antiderivative of cosec x is mathematically writen as ∫ cosec x dx. Multiply and divide cosec x by (cosec x  cot x), we get, ∫ cosec x dx = ∫ cosec x · (cosec x  cot x) / (cosec x  cot x) dx = ∫ (cosec^{2}x  cosec x cot x) / (cosec x  cot x) dx. By assuming cosec x  cot x = u. Then (cosec x cot x + cosec^{2}x) dx = du. Then the above integral becomes ∫ cosec x dx = ∫ du / u = ln u + C = ln cosec x  cot x + C.
What is the Integral of Csc x Cot x?
The derivative of csc x is csc x cot x. From this, the derivative of csc x is csc x cot x. Thus, ∫ csc x cot x dx =  csc x + C.
What is the Integration of Cosec^2x?
We know that the derivative of cot x is cosec^{2} x. So the derivative of cot x is cosec^{2}x. Thus, the integral of cosec^{2}x is cot x. i.e., ∫ cosec^{2}x dx = cot x + C.
How to Find the Integral of Cosec x by Substitution?
We can write ∫ cosec x dx as ∫ cosec x dx = ∫ cosec x · (cosec x  cot x) / (cosec x  cot x) dx. Now assume cosec x  cot x = u. Differentiating, (cosec x cot x + cosec^{2}x)dx = du. Then, ∫ cosec x dx = ∫ du / u = ln u + C = ln cosec x  cot x + C.
What is the Integral of cosec^{2}x cot x?
We can write ∫ cosec^{2}x cot x dx as ∫ cosec x (cosec x cot x) dx. Assume that cosec x = u then cosec x cot x dx = du. Then the above integral becomes, ∫ u du = u^{2}/2 + C = (cosec^{2}x)/2 + C.
What is the Integral of Cosec x from 0 to π/4?
We know that the integral of cosecantx is ln cosec x  cot x. Applying the limits 0 and π/4, we get ∫₀^{π/4 }cosec x dx = ln cosec π/4  cot π/4  ln cosec 0  cot 0 = diverges as cosec 0 is undefined.
visual curriculum