Step-by-Step Guide | Finding the Integral of cot(x) with Respect to x Using Substitution

∫cotxdx

To find the integral of cot(x) with respect to x, we can use a method called substitution

To find the integral of cot(x) with respect to x, we can use a method called substitution.

Let’s start by considering the integral:

∫cot(x)dx

The integral of cot(x) cannot be found directly, so we can rewrite cot(x) as 1/tan(x):

∫1/tan(x)dx

Next, we can rewrite tan(x) as sin(x)/cos(x):

∫1/(sin(x)/cos(x))dx

To simplify this expression further, we can multiply the integral by cos(x)/cos(x) to get rid of the fraction:

∫cos(x)/(sin(x))dx

Now, let’s introduce a new variable, u, which equals sin(x):

u = sin(x)

Then, we can rewrite the expression in terms of u:

∫cos(x)/(sin(x))dx = ∫1/u du

The integral of 1/u is ln|u|. Therefore, the integral becomes:

∫1/u du = ln|u| + C

Finally, substitute back u = sin(x) to get the final result:

ln|sin(x)| + C

So, the integral of cot(x) with respect to x is ln|sin(x)| + C, where C is the constant of integration.

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