Final answer:
The integral of sec(x)tan(x) from 0 to π/4 evaluates to ln|sec(0)| - ln|sec(π/4)| by taking the antiderivative of sec(x) and evaluating the resulting function at the upper and lower limits of integration.
Step-by-step explanation:
The integral of sec(x)tan(x) from 0 to π/4 can be solved by recognizing that the derivative of sec(x) is sec(x)tan(x). Therefore, the integral becomes the antiderivative of sec(x), which is ln|sec(x) + tan(x)|. We evaluate this at the bounds 0 and π/4.
At x = π/4, sec(π/4) = √2 and tan(π/4) = 1, so ln|sec(π/4) + tan(π/4)| is ln|√2 + 1|. At x = 0, sec(0) = 1 and tan(0) = 0, so ln|sec(0) + tan(0)| is ln|1 + 0| = ln|1|. The difference between these two values gives us Option (d): ln|sec(0)| - ln|sec(π/4)| = ln|1| - ln|√2|.