Question Detail

The reduction formula for ∫cosecⁿx dx is derived using integration by parts. Let V_n = ∫cosecⁿx dx.Choose u = cosecⁿ⁻²x and dv = cosec²x dx.Then, du = (n-2)cosecⁿ⁻³x (-cosec x cot x) dx = -(n-2)cosecⁿ⁻²x cot x dx.And v = ∫cosec²x dx = -cot x.Applying integration by parts (∫u dv = uv - ∫v du):V_n = (cosecⁿ⁻²x)(-cot x) - ∫(-cot x) [-(n-2)cosecⁿ⁻²x cot x] dxV_n = -cosecⁿ⁻²x cot x - (n-2)∫cosecⁿ⁻²x cot²x dxUse the identity cot²x = cosec²x - 1:V_n = -cosecⁿ⁻²x cot x - (n-2)∫cosecⁿ⁻²x (cosec²x - 1) dxV_n = -cosecⁿ⁻²x cot x - (n-2)∫(cosecⁿx - cosecⁿ⁻²x) dxV_n = -cosecⁿ⁻²x cot x - (n-2)∫cosecⁿx dx + (n-2)∫cosecⁿ⁻²x dxV_n = -cosecⁿ⁻²x cot x - (n-2)V_n + (n-2)V_(n-2)Rearrange to solve for V_n:V_n + (n-2)V_n = -cosecⁿ⁻²x cot x + (n-2)V_(n-2)(1 + n - 2)V_n = -cosecⁿ⁻²x cot x + (n-2)V_(n-2)(n-1)V_n = -cosecⁿ⁻²x cot x + (n-2)V_(n-2)V_n = -1/(n-1) cosecⁿ⁻²x cot x + (n-2)/(n-1) V_(n-2).This matches option A.