Question

Assertion (A): \( I_n = \int \cot^n x \, dx \), then \( I_6 + I_4 = -\frac{\cot^5 x}{5} \)
Reason (R): \( \int \cot^n x \, dx = \frac{-\cot^{n-1} x}{n - 1} - \int \cot^{n - 2} x \, dx \)

• A. A is false, R is false
• B. A is true, R is true
• C. A is true, R is false
• D. A is false, R is true

Hint:

Always double-check signs in reduction formulas — they are easy to misstate.

Answer 1

The Correct Option is C

The standard reduction formula for \( \int \cot^n x \, dx \) is:
\[ I_n = \frac{-\cot^{n-1}x}{n-1} + \int \cot^{n-2} x \, dx \] The assertion is correct: using this formula recursively, we get:
\[ I_6 + I_4 = -\frac{\cot^5 x}{5} \] But the reason is stated with the wrong sign: it says subtract instead of add the integral term.
Therefore, Assertion is true but Reason is false.