Question

find the integral of cotx using the substitution cotx = \frac{\cos x}{\sin x}. the integral of cotx is \square.

Explanation:

Step1: Rewrite cotx

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

Step2: Substitution

Let \(u = \sin x\), then \(\mathrm{d}u=\cos x\mathrm{d}x\). Substituting into the integral, we get \(\int\frac{\cos x}{\sin x}\mathrm{d}x=\int\frac{1}{u}\mathrm{d}u\).

Step3: Integrate 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 u

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)