Question Detail
Question
If I_n = ∫ cotⁿ x dx, the reduction formula for I_n is:
Correct Answer
Option C is the correct answer.
Detailed Explanation
To derive the reduction formula for ∫ cotⁿ x dx:I_n = ∫ cotⁿ x dx = ∫ cotⁿ⁻² x cot² x dxUse the identity cot² x = csc² x - 1:I_n = ∫ cotⁿ⁻² x (csc² x - 1) dxI_n = ∫ cotⁿ⁻² x csc² x dx - ∫ cotⁿ⁻² x dxThe second term is Iₙ₋₂. For the first term, let u = cot x, then du = -csc² x dx.So, ∫ cotⁿ⁻² x csc² x dx = ∫ uⁿ⁻² (-du) = -∫ uⁿ⁻² du= - (uⁿ⁻¹ / (n-1)) = - (cotⁿ⁻¹ x / (n-1)).Therefore, the reduction formula is: I_n = -cotⁿ⁻¹ x / (n-1) - Iₙ₋₂.
Hint
For reduction formulas involving powers of cot x, use the identity cot² x = csc² x - 1 and then a suitable substitution for the first integral.