Integral of Cot2x
The integral of cot2x is equal to (1/2) ln sin2x + C, where C is the integration constant. We can calculate the integration cot2x using the usubstitution method. Mathematically, we can write the integral of cot2x as ∫cot2x dx = (1/2) ln sin2x + C. Also, we know that cot2x can be expressed as the ratio of cos2x and sin2x. Therefore, after expressing cot in terms of sin and cos, we can determine the integral of cot2x using the quotient rule of integration.
In this article, we will explore the integral of cot2x, determine its formula using different methods of integration. We will also evaluate the integral of cot square x using integration formulas. In this article, we will solve different examples related to the concept for a better understanding.
What is the Integral of Cot2x?
The integral of cot2x is equal to (1/2) ln sin2x + C, where C is the integration constant. This formula of integration of cot2x can be determined using different integration formulas including the usubstitution method and expressing cot2x as the ratio of cos2x and sin2x. In the next section, let us go through the formula for the integral of cot2x.
Integral of Cot2x Formula
The formula for the integral of cot2x is ∫cot2x dx = (1/2) ln sin2x + C, where C is the constant of integration. This formula for the integration of cot2x can be used to determine its definite integral with different limits. The image given below shows the formula for the integral of cot2x:
Integral of Cot2x Using USubstitution
Now, we will prove that the integral of cot2x is equal to (1/2) ln sin2x + C using the usubstitution method of integration. For a function f(x) = cot2x, assume 2x = u. Differentiating both sides we have, 2 dx = du which implies dx = du/2. Also, we know that the integral of cotx is equal to ln sin x + K, where K is the integration constant. Using this formula, we have
∫cot2x dx = (1/2) ∫cot u du
= (1/2) ( ln sin u + K)
= (1/2) ln sin u + (1/2)K
= (1/2) lnsin 2x + C, where C = (1/2)K is the integration constant.
Hence, we have proved that the integral of cot2x is equal to (1/2) ln sin2x + C.
Integration of Cot2x Using Sin and Cos
We know that the cotangent function can be expressed as the ratio of the cosine function and the sine function. This implies cot2x can written as cot2x = cos2x/sin2x. Using this formula and the substitution method of integration, we can determine the integral of cot2x. Therefore, we have
∫cot2x dx = ∫(cos2x/sin2x) dx  (1)
Now, assume sin2x = t. Differentiating both sides, we have 2 cos2x dx = dt which implies cos2x dx = dt/2. Substituting these values in (1), we have
∫cot2x dx = ∫(1/t) (dt/2)
= (1/2) ∫(1/t) dt
= (1/2) ( ln t + K)
= (1/2) ln sin2x + (1/2)K
= (1/2) ln sin2x + C, where C = (1/2)K is the integration constant.
Integral of Cot Square x
Next, in this article, we will evaluate the integral of cot square x. We know the trigonometric identity 1 + cot^{2}x = cosec^{2}x which implies cot^{2}x = cosec^{2}x 1. Using this formula of trigonometry, we can evaluate the integral of cot^2x. Also, we know that the derivative of cotx is cosec^{2}x which implies the integral of cosec^{2}x is equal to cotx + K. Using these facts and formulas of trigonometry, we have
∫cot^2x dx = ∫cot^{2}x dx
= ∫(cosec^{2}x 1) dx
= ∫cosec^{2}x dx  ∫1 dx
= cotx  x + C
Hence, the integral of cot^2x is equal to cotx  x + C, where C is the integration constant.
Important Notes on Integral of Cot2x
 The integral of cot2x is given by, ∫cot2x dx = (1/2) ln sin2x + C, where C is the constant of integration.
 The integration of cot2x can be evaluated using the usubstitution method and using sin and cos.
 The integral of cot^2x is equal to cotx  x + C.
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Integral of Cot2x Examples

Example 1: What is the integral of cot2x tan2x?
Solution: We know that cot2x and tan2x are trigonometric reciprocals of each other which implies cot2x. = 1/tan2x and tan2x = 1/cot2x. Therefore, we can write cot2x tan2x = cot2x × 1/cot2x = 1. So, we have
∫cot2x tan2x dx = ∫1 dx
= x + C
Answer: Therefore, the integral of cot2x tan2x is equal to x + C.

Example 2: Evaluate the integral of cot2x plus tan2x whole square, that is, (tan2x + cot2x)^{2}.
Solution: To find the integral of (tan2x + cot2x)^{2}, we will use (A+B)^{2} = A^{2} + B^{2} + 2AB.
∫(tan2x + cot2x)^{2} dx = ∫ (tan^{2}2x + cot^{2}2x + 2tan2x cot2x) dx
= ∫ (tan^{2}2x + cot^{2}2x + 2) dx  [Because tan2x and cot2x are reciprocals of each other, therefore cot2x tan2x = 1]
= ∫[(sec^{2}2x  1) + (cosec^{2}2x  1) + 2] dx  [Using trigonometric identity 1 + tan^{2}x = sec^{2}x and 1 + cot^{2}x = cosec^{2}x]
= ∫(sec^{2}2x  1 + cosec^{2}2x  1 + 2) dx
= ∫sec^{2}2x dx + ∫cosec^{2}2x dx
= (1/2) tan2x  (1/2) cot2x + C
Answer: The integral of (tan2x + cot2x)^{2} is (1/2) tan2x  (1/2) cot2x + C
FAQs on Integral of Cot2x
What is Integral of Cot2x in Calculus?
The integral of cot2x is equal to (1/2) ln sin2x + C, where C is the integration constant. We can calculate the integration cot2x using the usubstitution method. Mathematically, we can write the integral of cot2x as ∫cot2x dx = (1/2) ln sin2x + C.
How To Find the Integral of Cot2x?
The formula of the integration of cot2x can be determined using different integration formulas including the usubstitution method and expressing cot2x as the ratio of cos2x and sin2x.
What is the Integral of Cot Square x?
The integral of cot^2x is equal to cotx  x + C, where C is the integration constant. We can find this integral of cot square x using the trigonometric identity 1 + cot^{2}x = cosec^{2}x.
Is the Antiderivative of Cot2x the Same as the Integral of Cot2x?
Antiderivative of a function is nothing but its integral as integration is the reverse process of differentiation (also known as antidifferentiation).
What is the Formula for The Integration of Cot2x?
The formula for the integration of cot2x is given by, ∫cot2x dx = (1/2) ln sin2x + C, where C is the constant of integration.
What is the Derivative of Cot2x?
The derivative of cot2x is given by, d(cot2x)/dx = 2 cosec^{2}2x. We know the fact that the derivative of cotx is equal to cosec^{2}x. Using this formula and integration rules, we can determine the derivative of cot2x.
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