$$ \int2\cosec\text{xcotx}-\sec^2x\text{ dx} $$

We are given the integral:

\[
\int 2 \csc(x) \cot(x) - \sec^2(x) \, dx
\]

### Step 1: Break the integral into two parts.

\[
\int 2 \csc(x) \cot(x) \, dx - \int \sec^2(x) \, dx
\]

### Step 2: Solve each integral separately.

#### Integral 1: $ \int 2 \csc(x) \cot(x) \, dx $

We know that the derivative of $ \csc(x) $ is $ -\csc(x) \cot(x) $. Therefore,

\[
\int 2 \csc(x) \cot(x) \, dx = -2 \csc(x)
\]

#### Integral 2: $ \int \sec^2(x) \, dx $

We know that the derivative of $ \tan(x) $ is $ \sec^2(x) $. Therefore,

\[
\int \sec^2(x) \, dx = \tan(x)
\]

### Step 3: Combine the results.

Now, putting both integrals together:

\[
-2 \csc(x) - \tan(x) + C
\]

where $ C $ is the constant of integration.

### Final Answer:
\[
\boxed{-2 \csc(x) - \tan(x) + C}
\]

Would you like more details on any of these steps or have further questions?

Here are 5 related questions to deepen your understanding:
1. What is the derivative of $ \csc(x) \cot(x) $?
2. How is the integral of $ \sec^2(x) $ derived?
3. What is the geometric meaning of $ \csc(x) $ and $ \cot(x) $?
4. Can you compute $ \int \sec(x) \, dx $?
5. How would you approach $ \int \csc^2(x) \, dx $?

**Tip:** Knowing basic trigonometric integrals, like $ \sec^2(x) $ and $ \csc(x) \cot(x) $, can speed up your solving process significantly!

$$ \int 2\cosec(x)\cot(x) - \sec^2(x) \, dx $$

Learn how to compute the integral of 2csc(x)cot(x) - sec^2(x) with step-by-step solutions. This trigonometric integration covers key calculus concepts such as the integral of sec^2(x) and the derivative of csc(x)cot(x). Suitable for high school and college-level calculus students.

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