A Step-by-Step Guide to Finding the Integral of Cot(x) Using Integration by Substitution

∫cotxdx

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: Start by writing down the integral:

∫cot(x) dx

Step 2: Recall the trigonometric identity cot(x) = cos(x) / sin(x). You can rewrite the integral as:

∫(cos(x) / sin(x)) dx

Step 3: Now, let’s substitute u = sin(x). This means that du/dx = cos(x) (derivative of sin(x)), or dx = du / cos(x). Substitute these values into the integral:

∫(1/sin(x)) (du/cos(x))

Simplifying this expression gives:

∫(1/u) du

Step 4: The integral has now become much simpler. We can rewrite it as:

ln|u| + C

where C is the constant of integration.

Step 5: Substitute back u = sin(x) to get the final result:

ln|sin(x)| + C

So, the integral ∫cot(x) dx is equal to ln|sin(x)| + C.

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