Solving the Integral of cot(x): Step-by-Step Explanation and Solution using Trigonometric Identities

cot(x)dx

To integrate cot(x)dx, we can use a trigonometric identity that states:

cot(x) = 1/tan(x)

So, we can rewrite the integral as:

∫ cot(x)dx = ∫(1/tan(x))dx

Next, we can substitute tan(x) with its equivalent form of sin(x)/cos(x):

∫(1/(sin(x)/cos(x)))dx

Simplifying further, we get:

∫(cos(x)/sin(x))dx

To solve this integral, we can use a u-substitution

To integrate cot(x)dx, we can use a trigonometric identity that states:

cot(x) = 1/tan(x)

So, we can rewrite the integral as:

∫ cot(x)dx = ∫(1/tan(x))dx

Next, we can substitute tan(x) with its equivalent form of sin(x)/cos(x):

∫(1/(sin(x)/cos(x)))dx

Simplifying further, we get:

∫(cos(x)/sin(x))dx

To solve this integral, we can use a u-substitution. Let u = sin(x), then du/dx = cos(x) dx. Rearrange this equation to solve for dx: dx = du/cos(x).

Substituting these values into the integral, we get:

∫(1/u)(du/cos(x))

Now, we can cancel out the cos(x) terms:

∫(1/u)du

This integral can be solved easily:

∫(1/u)du = ln|u| + C

Substituting u back in terms of x, we have:

ln|sin(x)| + C

Therefore, the final answer to ∫ cot(x)dx is ln|sin(x)| + C, where C is the constant of integration.

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