Question

Evaluate the integral: \[ \int \cot^2(x) \, dx \]

• A. \( \cot(x) + x + k \)
• B. \( -\cot(x) + x + k \)
• C. \( -\cot(x) - x + k \)
• D. \( \cot(x) - x + k \)

Hint:

Use the identity \( \cot^2(x) = \csc^2(x) - 1 \) to simplify integrals involving \( \cot^2(x) \). Then, split the integral and integrate each term.

Answer 1

The Correct Option is C

We know that \( \cot^2(x) = \csc^2(x) - 1 \) (using the trigonometric identity). So, the integral becomes: \[ \int \cot^2(x) \, dx = \int (\csc^2(x) - 1) \, dx \] Now, split the integral: \[ = \int \csc^2(x) \, dx - \int 1 \, dx \] We know that: \[ \int \csc^2(x) \, dx = -\cot(x) \] and \[ \int 1 \, dx = x \] Therefore, the integral is: \[ -\cot(x) - x + k \] Thus, the correct answer is: \[ \boxed{-\cot(x) - x + k} \]