How to Find the Integral of Cot(x) and its Substitution Technique

∫cot(x)dx

To find the integral of cot(x), we can use a technique called integration by substitution

To find the integral of cot(x), we can use a technique called integration by substitution. Let’s go through the steps:

Step 1: Identify the substitution.
To begin, take a look at the integrand cot(x). The derivative of cot(x) is -csc^2(x). So, let’s choose u = sin(x) as our substitution.

Step 2: Differentiate the substitution.
We have u = sin(x), and we need to find du/dx. Taking the derivative of u with respect to x gives us du/dx = cos(x).

Step 3: Rearrange the derivative equation.
We have du/dx = cos(x). Rearranging this equation, we get dx = du/cos(x).

Step 4: Substitute and rewrite the integral.
Now, substitute u = sin(x) and dx = du/cos(x) into our original integral:
∫cot(x)dx = ∫cot(x) * (du/cos(x))

Step 5: Simplify the integral.
Since cos(x)/cos(x) equals 1, we can simplify the integral further:
∫cot(x)dx = ∫cot(x) * (du/cos(x))
= ∫cot(x) * du

Step 6: Evaluate the integral.
The integral of cot(x) with respect to u is ln|sin(x)| + C, where C is the constant of integration.

Step 7: Convert back to the original variable.
Remember that we used u = sin(x) as a substitution. Converting back to the original variable, we get:
∫cot(x)dx = ln|sin(x)| + C

So, the antiderivative of cot(x) is ln|sin(x)| + C, where C is the constant of integration.

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