$$ \int ( \cos x - \sin x ) \cdot \operatorname { cosec } x \cdot d x $$

Question

UpStudy AI · Accepted Answer

To evaluate the integral:

$$
\int (\cos x - \sin x) \cdot \csc x \, dx
$$

Let's simplify the integrand step by step.

1. **Rewrite $$\csc x$$:**
   
   $$\csc x = \frac{1}{\sin x}$$, so the integral becomes:

$$
   \int \frac{\cos x - \sin x}{\sin x} \, dx
   $$

2. **Separate the terms:**

$$
   \int \left( \frac{\cos x}{\sin x} - \frac{\sin x}{\sin x} \right) dx = \int \cot x \, dx - \int 1 \, dx
   $$

3. **Integrate each term:**

- The integral of $$\cot x$$ is $$\ln|\sin x|$$.
   - The integral of $$1$$ is $$x$$.

4. **Combine the results:**

$$
   \ln|\sin x| - x + C
   $$

where $$C$$ is the constant of integration.

**Final Answer:**

$$
\boxed{\ln\,|\sin x| \ -\ x\ +\ C}
$$
The integral simplifies to $$ \ln|\sin x| - x + C $$.

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