Integrating Cot²(x) Using Trigonometric Identities: Step-by-Step Explanation and Solution

∫cot²(x)dx

To evaluate the integral ∫cot²(x)dx, we can use integration techniques and trigonometric identities

To evaluate the integral ∫cot²(x)dx, we can use integration techniques and trigonometric identities.

First, let’s begin by applying the Pythagorean identity for cotangent:
1 + cot²(x) = csc²(x)

Rearranging the equation, we get:
cot²(x) = csc²(x) – 1

This allows us to express the integral as:
∫cot²(x)dx = ∫(csc²(x) – 1) dx

Now, we can split the integral into two separate integrals:
∫csc²(x)dx – ∫1 dx

Let’s evaluate these integrals one by one:

1) Integral of csc²(x)dx:
To solve this integral, let’s use the identity:
∫csc²(x)dx = -cot(x)

Therefore,
∫csc²(x)dx = -cot(x) + C
(where C is the constant of integration)

2) Integral of 1 dx:
The integral of a constant is simply the constant multiplied by x. Hence,
∫1 dx = x

Now, let’s put everything together:

∫cot²(x)dx = ∫(csc²(x) – 1) dx
= ∫csc²(x)dx – ∫1 dx
= (-cot(x) + C) – x + C
= -cot(x) – x + 2C

Therefore, the final answer to the integral ∫cot²(x)dx is:
∫cot²(x)dx = -cot(x) – x + 2C
(where C is the constant of integration)

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