Mastering the Integral of Cot(x) Using the Substitution Method

∫cotxdx

To evaluate the integral ∫cot(x) dx, we can use the substitution method

To evaluate the integral ∫cot(x) dx, we can use the substitution method.

Let’s start by considering the definition of the cotangent function:

cot(x) = cos(x)/sin(x)

Now, let’s make the substitution u = sin(x). This implies that du/dx = cos(x), or equivalently, dx = du/cos(x).

Substituting these expressions into the integral, we have:

∫cot(x) dx = ∫(cos(x)/sin(x)) dx
= ∫(cos(x)/(u)) (du/cos (x))

Notice that the cos(x) terms cancel out, leaving us with:

∫(1/u) du

Now, we can proceed to integrate:

∫(1/u) du = ln|u| + C

Finally, substituting back u = sin(x), we get:

∫cot(x) dx = ln|sin(x)| + C

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

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