Integral of Cot x is ln |sin x| + C. Cot x is among one of the trigonometric functions that is the ratio of cosine and sine. The integral of cot x is mathematically represented as ∫cot x dx = ln |sinx| + C.

In this article, we will explore the integral of cot x, the integral of cot x formula, the derivation of the integral of cot x, definite integral of cot x along with some examples based on the integral of cot x.

What is 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. The integral of cot x means finding the antiderivative of the cot x. The process of finding the anti-derivative of a function is called the integration. The result of the integration is called as integral. Hence, the antiderivative of the cot x is ln |sin x| + C.

Read in Detail:

• Calculus in Maths
• Integral Calculus

Integral of Cot x Formula

The integral of cot x formula is given by:

∫cot x dx = ln |sin x| + C

Integral of Cot x in Terms of Cosec x

Integral of Cot x in terms of cosec x is given as follows:

∫cot x dx = – ln |cosec x| + C

Integral of Cot x Proof

We can derive the integral of cot x by using the Substitution Method in integration.

Integral of Cot x by Substitution Method

To prove integral of cot x we will use the integration by substitution method which is described below:

We know that,

cot x = cos x / sin x

Integrating both sides we get,

∫cot x dx = ∫ [cos x / sin x] dx —-(1)

Let t = sin x

Differentiating both sides w.r.t t, we get

dt = cos x dx

Putting the above values in equation (1)

∫cot x dx = ∫ [1 / t] dt

∫cot x dx = ln |t| + C

Putting value of t

∫cot x dx = ln |sin x| + C

The integral of cot x is ln |sin x| + C.

Definite Integral of Cot x dx 

Integral of cot x with the upper and lower limit is called as the definite integral of cot x. In this we apply the limits and evaluate the resultant value for the integral. The value of definite integral of cot x is given below:

Integral of Cot x from 0 to pi/2 

The value of integral of cot x with lower limit 0 and upper limit π/2 is given below:

We know that,

∫cot x dx = ln |sin x| + C

Applying lower limit = 0 and upper limit = π/2, we get

∫₀^π/2cot x dx = [ln |sin x| ]₀^π/2

∫₀^π/2cot x dx = ln |sin(π/2) | – |ln sin (0) |

∫₀^π/2cot x dx = ln |sin(π/2) | – |ln 0|

Since, the ln 0 is not defined, the definite integral ∫₀^π/2cot x dx diverges.

Integral of Cot x from pi/4 to pi/2 

The value of integral of cot x with lower limit π/4 and upper limit π/2 is given below:

We know that,

∫cot x dx = ln |sin x| + C

Applying lower limit = π/4 and upper limit = π/2

∫_π/4^π/2cot x dx = [ln |sin x| ]_π/4^π/2

⇒ ∫_π/4^π/2cot x dx = ln |sin(π/2) | – |ln sin(π/4) |

⇒ ∫_π/4^π/2cot x dx = ln 1 – ln (1/√2)

⇒ ∫_π/4^π/2cot x dx = ln 1 – [ln 1 – ln √2]

⇒ ∫_π/4^π/2cot x dx = ln (√2)

Integral of Cot x from pi/4 to pi/2 is ln (√2).

Important Notes

Some important points related to integral of cot x are:

• ∫cot x dx = ln |sinx| + C

• ∫cot x dx = ln |cosec x|^-1 + C [As sinx = (cosec x)^-1]

• The definite integral of cot x diverges when the upper limit is pi/2 and lower limit is 0.

• The definite integral of cot x from upper limit pi/2 to lower limit pi/4 evaluates to ln (√2).

• ∫cot² x dx = – cosec x + C

Read More:

• Integration Formulas
• Integration of Trigonometric Functions
• Integration of Tan x
• Integration of Cos x
• Integration of Sec x

Solved Examples on Integral of Cot x

Example 1: Find ∫cot 6x dx

Solution:

We have ∫cot 6x dx ——(1)

Let t = 6x

Differentiating w.r.t t

dt = 6 dx

⇒ dx = dt / 6

Putting in (1)

∫cot 6x dx = ∫cot t (dt / 6)

⇒ ∫cot 6x dx = (1 / 6) ∫cot t dt

⇒ ∫cot 6x dx = (1 / 6) [ln |sin t| + C]

⇒ ∫cot 6x dx = (1 / 6) [ln |sin (6x) | + C]

Example 2: Evaluate: ∫cot x cosec²x dx

Solution:

Let I = ∫cot x cosec²x dx —–(1)

Take t = cot x

Differentiating w.r.t t

dt = – cosec²x dx

putting in (1)

I = -∫t dt

⇒ I = -t² / 2 + C (putting values)

⇒ I = – cot²x / 2 + C

⇒ ∫cot x cosec²x dx = – cot²x / 2 + C

Example 3: Solve ∫cot x. sec x dx

Solution:

I = ∫cot x. sec x dx

We know that,

cot x = cos x / sin x and sec x = 1 / cos x

Putting in I

I = ∫ [cos x / sin x]. [1 / cos x] dx

⇒ I = ∫ [1 / sin x] dx

⇒ I = ∫ cosec x dx

⇒ I = – ln | cosec x + cot x| + C

Example 4: Evaluate ∫cot² x dx

Solution:

I = ∫cot² x dx

We know that,

[d / dx] (cosec x) = – cot² x

cot² x = – [d / dx] (cosec x)

Putting in I

I = ∫ – [d / dx] (cosec x) dx

By the property ∫[d / dx] f(x) dx = f(x) + C

I = – cosec x + C

Practice Questions on Integral of Cot x

Q1. Solve ∫cot x. cos x dx.

Q2. Evaluate the integral ∫ [cot x / √ (6 + 16cot²x)] dx.

Q3. Find ∫ cot (4x) dx.

Q4. Evaluate ∫ (1 + cot x) / (1 – cot x) dx

Integral of Cot x – FAQs

What is the antiderivative of cot x?

The antiderivative of cot x is ln |sin x| + C.

How to prove the Integral of Cot x?

We can prove the integral of cot x by applying the substitution method.

Is the derivative of cot x is equal to the integral of cot x?

No, the derivative of cot x is not equal to the integral of cot x. The derivative of cot x = -cosec² x whereas the integral of cot x = ln |sinx| + C.

What is the formula of integral of cot x?

The formula for the integral of cot x is given by:

∫cot x dx = ln |sin x| + C

What is the value of definite integral of cot x in the interval pi/4 to pi/2?

The value of the definite integral of cot x in the interval pi/4 to pi/2 is ln √2.

What is differentiation of cot X?

The differentiation of cot x is -cosec² x

What is the integral of cot² x?

The integral of cot² x is – cosec x + C.

What is integral of cot x dx?

The integral of cot x dx is ln |sin x| + C