How to Find the Integral of cot²(x) With Respect to x | A Step-by-Step Guide

∫cot²(x)dx

To find the integral of cot²(x) with respect to x, we can use trigonometric identities and integration techniques

To find the integral of cot²(x) with respect to x, we can use trigonometric identities and integration techniques.

First, let’s rewrite cot²(x) using trigonometric identities. The identity we can use is:

cot²(x) = csc²(x) – 1

So, the integral becomes:

∫cot²(x)dx = ∫(csc²(x) – 1)dx

Now, let’s integrate both terms separately.

∫csc²(x)dx:
To integrate csc²(x), we can use the identity:

csc²(x) = cot²(x) + 1

So, the integral becomes:

∫csc²(x)dx = ∫(cot²(x) + 1)dx

Since we already know that ∫cot²(x)dx = ∫(csc²(x) – 1)dx, we can rewrite the integral as:

∫csc²(x)dx = ∫(cot²(x) + 1)dx = ∫cot²(x)dx + ∫1 dx

Now, let’s integrate the second term:

∫1 dx:
Integrating 1 with respect to x gives us x. So,

∫1 dx = x

Now, let’s go back to the expression:

∫cot²(x)dx + ∫1 dx

We can substitute the integral of cot²(x) with ∫(csc²(x) – 1)dx:

∫(csc²(x) – 1)dx + ∫1 dx

Simplifying, we get:

∫cot²(x)dx = ∫csc²(x)dx – ∫1 dx + ∫1 dx
∫cot²(x)dx = ∫csc²(x)dx

Now, we can integrate ∫csc²(x)dx:

∫csc²(x)dx:
To integrate csc²(x), we can use the substitution method. Let’s substitute u = cot(x) and see what happens.

Differentiating both sides with respect to x:

du/dx = -csc²(x)
dx = -du/csc²(x)

Substituting into the integral:

∫csc²(x)dx = ∫csc²(x) (-du/csc²(x))
∫csc²(x)dx = -∫du
∫csc²(x)dx = -u + C

Substituting back u = cot(x), we get:

∫csc²(x)dx = -cot(x) + C

Hence, the integral of cot²(x)dx is:

∫cot²(x)dx = -cot(x) + C

Where C represents the constant of integration.

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