Question

Calculati ( explicit va rog )$\int\limits^a_b {\frac{1}{sin^{2}x * cos^{2}x \ } } \, dx$

Answer 1

I = ∫ 1/(sin²x•cos²x) dx

I = ∫ 4/(4sin²x•cos²x) dx

I = ∫ 4/sin²(2x) dx

I = 4•∫ 1/sin²(2x) dx

Știm că: [ctg(t)]' = -1/sin²(t)

I = 4•(-1/2)•∫ [ctg(2x)]' dx

I = -2•ctg(2x) dx

∫ₐᵇ 1/(sin²x•cos²x) dx =

= -2•ctg(2b) + 2•ctg(2b) =

= 2•[ctg(2a) - ctg(2b)]