Master the Integration of Cot²(x) – Key Steps and Formulas for Success

∫cot²(x)dx

To find the integral of cot²(x), we can use a trigonometric identity and a basic integration technique

To find the integral of cot²(x), we can use a trigonometric identity and a basic integration technique.

First, let’s recall the trigonometric identity for cotangent squared:

cot²(x) = 1 + csc²(x)

By substituting this identity into the integral, we get:

∫cot²(x)dx = ∫(1 + csc²(x))dx

Next, we can split this integral into two separate integrals:

∫(1 + csc²(x))dx = ∫1dx + ∫csc²(x)dx

The integral of 1dx is simply x:

∫1dx = x + C1, where C1 is the constant of integration for the first integral.

Now, let’s focus on the second integral, which is the integral of csc²(x). To solve this integral, we can use a u-substitution.

Let u = cot(x), then du = -csc²(x)dx

Rearrange the equation to solve for dx:

dx = -du/csc²(x)

Substituting du and -du/csc²(x) for dx in the integral, we get:

∫csc²(x)dx = ∫csc²(x)(-du/csc²(x))

The csc²(x) terms cancel out, leaving us with:

∫csc²(x)(-du/csc²(x)) = -∫du

The integral of -du is simply -u:

-∫du = -u + C2, where C2 is the constant of integration for the second integral.

Now, let’s put everything together:

∫cot²(x)dx = x – cot(x) + C, where C = C1 + C2 is the constant of integration for the entire expression.

Therefore, the integral of cot²(x) is x – cot(x) + C, where C is a constant.

More Answers: