Final answer:
To evaluate the integral of 1/(1+cosx), we use a trigonometric identity to rewrite the expression, apply a substitution, and then integrate, resulting in the antiderivative tan(x/2) + C.
Step-by-step explanation:
The question asks to evaluate the integral with respect to x of the expression (1)/(1+cosx). Evaluating this integral involves a trigonometric identity and substitution method. First, we use the identity cos(2x) = 2cos2(x) - 1, which can be rewritten as cos2(x) = (1 + cos(2x))/2. The integrand 1/(1+cosx) can thus be rewritten as 1/(2cos2(x/2)), which is 1/2 sec2(x/2). Now, we can set u = x/2 and du = (1/2)dx, therefore dx = 2du. Substituting these into the integral gives us the integral of sec2(u) with respect to u which is simply tan(u) + C, or tan(x/2) + C after reverting back to the original variable x.