Question Detail
Question
Evaluate ∫0π/4 sec² x e4 tan x dx.
Correct Answer
Option D is the correct answer.
Detailed Explanation
This integral can be solved using u-substitution.Let u = 4 tan x.Then du = 4 sec² x dx, which means sec² x dx = (1/4) du.Now change the limits of integration:When x = 0, u = 4 tan(0) = 0.When x = π/4, u = 4 tan(π/4) = 4 * 1 = 4.The integral becomes:∫04 eu (1/4) du = (1/4) ∫04 eu du.Integrate eu: (1/4) [eu]04.Evaluate the limits: (1/4) (e⁴ - e0).Since e0 = 1, the result is (1/4)(e⁴ - 1).
Hint
Use u-substitution by letting u be the exponent of e. Don't forget to change the limits of integration.