Question Detail

This integral can be solved using a u-substitution.Let u = e2x - e-2x.Now, find du/dx:du/dx = d/dx(e2x) - d/dx(e-2x)Using the chain rule, d/dx(eax) = a eax:du/dx = (2e2x) - (-2e-2x)du/dx = 2e2x + 2e-2xdu/dx = 2(e2x + e-2x)So, du = 2(e2x + e-2x) dx.This means (e2x + e-2x) dx = (1/2)du.Substitute u and du into the integral:∫ (1/u) * (1/2) du = (1/2) ∫ (1/u) duNow, integrate with respect to u:= (1/2) ln|u| + CSubstitute u = e2x - e-2x back into the expression:= (1/2) ln|e2x - e-2x| + CThis matches option C.