设In=∫(tanx)^ndx,n>=2,证明：In=1\n-1(tanx)^n-1-In-2

Question

设In=∫(tanx)^ndx,n>=2,证明：In=1\n-1(tanx)^n-1 -In-2

Answer 1

I(n) = ∫ tanⁿx dx
= ∫ tanⁿ⁻²x·tan²x dx
= ∫ tanⁿ⁻²x·(sec²x - 1) dx
= ∫ tanⁿ⁻²x·sec²x dx - ∫ tanⁿ⁻²x dx
= ∫ tanⁿ⁻²x d(tanx) - I(n - 2)
I(n) = tanⁿ⁻¹x/(n - 1) - I(n - 2) for n ≥ 2