Integral of Cot x
Before going to learn what is the integral of cot x, let us recall few facts about the cot (or) cotangent function. In a right angled triangle, if x is one of the acute angles, then cot x is the ratio of the adjacent side of x to the opposite side of x. So it can be written as (cos x)/(sin x) as cos x = (adjacent)/(hypotenuse) and sin x = (opposite)/(hypotenuse). We use these facts to find the integral of cot x.
Let us learn the integral of cot x formula along with its proof and examples.
1.  What is the Integral of Cot x dx? 
2.  Integral of Cot x Proof by Substitution 
3.  Definite Integral of Cot x dx 
4.  FAQs on Integral of Cot x 
What is the Integral of Cot x dx?
The integral of cot x dx is equal to ln sin x + C. It is represented by ∫ cot x dx and so ∫ cot x dx = ln sin x + C, where 'C' is the integration constant. Here,
 '∫' is the symbol of integration.
 cot x is the integrand.
 dx represents that the integration is with respect to x.
We use the integration by substitution method to prove this integration of cot x formula. We will see it in the upcoming section.
Integral of Cot x Proof by Substitution
Here is the derivation of the formula of integral of cot x by using integration by substitution method. For this, let us recall that cot x = cos x/sin x. Then ∫ cot x dx becomes
∫ cot x dx = ∫ (cos x)/(sin x) dx
Substitute sin x = u. Then cos x dx = du. Then the above integral becomes
= ∫ (1/u) du
= ln u + C (Because ∫ 1/x dx = lnx + C)
Substitute u = sin x back here,
= ln sin x + C
Thus, ∫ cot x dx = ln sin x + C.
Hence proved.
Definite Integral of Cot x dx
The definite integral of cot x dx is an integral with lower and upper bounds. To evaluate it, we substitute the upper and lower limits in the value of the integral cot x dx and subtract them in the same order. We will work on some definite integrals of cot x dx.
Integral of Cot x From 0 to pi/2
∫\(_0^{\pi/2}\) cot x dx = ln sin x \(\left. \right_0^{\pi/2}\)
= ln sin π/2  ln sin 0
= ln 1  ln 0
We know that ln 1 = 0 and ln 0 is NOT defined. So
∫\(_0^{\pi/2}\) cot x dx = Diverges
Therefore, the integral of cot x from 0 to π/2 diverges.
Integral of Cot x From pi/4 to pi/2
∫\(_{\pi/4}^{\pi/2}\) cot x dx = ln sin x \(\left. \right_{\pi/4}^{\pi/2}\)
= ln sin π/2  ln sin π/4
= ln 1  ln 1/√2
By one of the properties of logarithms, ln (m/n) = ln m  ln n. So
= ln 1  ln 1 + ln √2
= ln √2
∫\(_{\pi/4}^{\pi/2}\) cot x dx = ln √2
Therefore, the integral of cot x from π/4 to π/2 is equal to ln √2.
Important Notes on Integral of Cot x dx:
 ∫ cot x dx = ln sin x + C
 Since sin x and csc x are reciprocals of each other,
∫ cot x dx = ln csc x^{1} + C =  ln csc x + C  ∫ cot^{2}x dx =  csc x + C as d/dx(csc x) = cot^{2}x + C
Topics Related to Integral of Cot x dx:
Examples on Integral of Cot x

Example 1: Evaluate the integral ∫ cot 3x dx.
Solution:
Assume that 3x = u then 3 dx = du (or) dx = du/3.
Then the above integral becomes,
∫ cot u (du/3) = (1/3) ln sin u + C
Substitute u = 3x back here,
∫ cot 3x dx = (1/3) ln sin 3x + C
Answer: The integral of cot 3x dx is (1/3) ln sin 3x + C.

Example 2: Evaluate the integral ∫ cot x csc x dx.
Solution:
∫ cot x csc x dx = ∫ (cos x)/(sin x) · (1/sin x) dx
= ∫ (cos x)/(sin^{2}x) dx
Let sin x = u. Then cos x dx = du.
= ∫ (1/u^{2}) du
= ∫ u^{2} du
= (u^{1})/(1) + C
= 1/u + C
= 1/(sin x) + C
=  csc x + C
Answer: The integral of cot x csc x is csc x + C.

Example 3: Evaluate the integral ∫ cot^{2}x dx.
Solution:
We know that d/dx (csc x) = cot^{2}x.
From this, d/dx (csc x) = cot^{2}x.
Since the integral is nothing but the antiderivative,
∫ cot^{2}x dx = csc x + C
Answer: The integral of cot^2x dx is csc x + C.
FAQs on Integral of Cot x
What is the Integral of Cot x?
The integral of cot x is ln sin x + C. It is mathematically denoted as ∫ cot x dx = ln sin x + C.
Is Derivative of Cot x Equal to Integral of Cot x?
No, the derivative of cot x is csc^{2}x and the integral of cot x is ln sin x + C. i.e.,
 d/dx(cot x) = csc^{2}x
 ∫ cot x dx = ln sin x + C
What is the Integral of Cot x From 0 to Pi?
We know that ∫ cot x dx = ln sin x + C. Substituting the limits, ln sin π  ln sin 0 = Diverges (as 0 is NOT in the domain of logarithmic function).
What is the Integral of Cot x Sec x?
∫ cot x sec x dx = ∫(cos x)/(sin x) · (1/cos x) dx = ∫ 1/sin x dx = ∫ csc x dx =  ln csc x + cot x + C.
How to Derive the Integral of Cot x Formula?
To derive the formula for ∫ cot x dx, we write cot x as (cos x)/(sin x). Then the integral becomes ∫ (cos x)/(sin x) dx. Assume that sin x = u, then cos x dx = du. Then the integral becomes ∫ (1/u) du = ln u + C (or) ln sin x) + C. Therefore, ∫ cot x dx = ln sin x + C.
What is the Integral of Cot^2x Equal to?
We know that the derivative of csc x is cot^{2}x. So the derivative of csc x is cot^{2}x. Since integration is the reverse process of differentiation, ∫ cot^{2}x dx = csc x + C.
What is the Integral of Cot x Csc^{2}x dx?
Assume that cot x = u then csc^{2}x dx = du. Then the given integral becomes ∫ u du = u^{2}/2 + C = cot^{2}x/2 + C.
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