Question Detail

When using the Weierstrass substitution t = tan(x/2), we have the identities:dx = 2 dt / (1 + t2) and cos x = (1 - t2) / (1 + t2).The integral is ∫ dx / (1 - cos x).However, if the question was intended to be ∫ dx / (1 + cos x), the solution matches option B. Let's assume the integral was ∫ dx / (1 + cos x) due to the provided options.Using the identity 1 + cos x = 2cos2(x/2):∫ dx / (1 + cos x) = ∫ dx / (2cos2(x/2)) = (1/2) ∫ sec2(x/2) dx.Let u = x/2, so du = (1/2) dx, which means dx = 2 du.Substitute these into the integral: (1/2) ∫ sec2(u) (2 du) = ∫ sec2(u) du.The integral of sec2(u) is tan(u).So, tan(u) + C. Substitute back u = x/2: tan(x/2) + C.This matches option B. If the integral was indeed ∫ dx / (1 - cos x), the result would be -cot(x/2) + C.