Question

In Exercises \(25-46,\) use substitution to evaluate the integral. $$\int \frac{d x}{\cot 3 x}$$

Short answer

The evaluated integral is \(-\frac{1}{3} \ln |\cot(3x)| + C\)

Step-by-step solution

Choice of Substitution

Choose a suitable substitution, in this case let \(u = \cot(3x)\). Then differentiate \(u\) with respect to \(x\) to find \(du/dx = -3 \csc^2(3x)\). We rearrange to find \(dx = -\frac{1}{3} \csc^2(3x) du\).

Change of Variables

Replace cotangent and \(dx\) in integral with new variables. The integral thus changes to \(\int -\frac{1}{3} \csc^2(3x) du\)

Evaluate the Integral

Now that the integral is simplified, its value can be found by simple integration which gives \(-\frac{1}{3} \int du\)

Back Substitute u

Finally, substitute back for \(u\), the integral is therefore \(-\frac{1}{3} \ln |u|\). Substituting \(u = \cot(3x)\) will give the final answer as \(-\frac{1}{3} \ln |\cot(3x)|+ C\), where \(C\) is the constant of integration.