Final answer:
To solve the integral using u-substitution, set u = tan(x), change the integration limits, and integrate the simpler expression, leading to a final answer of 17/6.
Step-by-step explanation:
The student has asked to solve by u-substitution the integral from 0 to arctan(2) of (sec(x))^2 / 4 + (tan(x))^2 dx. To approach this problem, we first notice that (sec(x))^2 and (tan(x))^2 are derivatives of each other. Recognizing tan(x) as a good candidate for u-substitution, we let u = tan(x). The derivative of u with respect to x is du/dx = (sec(x))^2. This allows us to rewrite the integral as ∑(1/4 + u^2) du.
Since we are integrating from 0 to arctan(2), we need to change the limits of integration in terms of u. When x = 0, u = tan(0) = 0. When x = arctan(2), u = tan(arctan(2)) = 2. Now we have ∑₀²(1/4 + u^2) du, which simplifies to the integral of 1/4 du plus the integral of u^2 du from 0 to 2.
These are basic integrals, resulting in u/4 + u^3/3 evaluated from 0 to 2. Plugging in the limits, we get (2/4 + 2^3/3) - (0/4 + 0^3/3), which simplifies to 8/3 + 1/2 or 17/6 as the final answer.