Question

find the integral of cotx using the substitution cotx = (cos x)/(sin x)
the integral of cotx is \boxed{}

Explanation:

Step1: Rewrite cotx

We know that \(\cot x=\frac{\cos x}{\sin x}\), so \(\int\cot xdx = \int\frac{\cos x}{\sin x}dx\).

Step2: Use substitution

Let \(u = \sin x\), then \(du=\cos xdx\). Substituting these into the integral, we get \(\int\frac{\cos x}{\sin x}dx=\int\frac{du}{u}\).

Step3: Integrate \(\frac{1}{u}\)

The integral of \(\frac{1}{u}\) with respect to \(u\) is \(\ln|u|+C\) (where \(C\) is the constant of integration).

Step4: Substitute back

Since \(u = \sin x\), we substitute back to get \(\ln|\sin x|+C\).

Answer:

\(\ln|\sin x| + C\) (where \(C\) is the constant of integration)