Question Detail

This integral can be solved using u-substitution.Let u = e2x - e-2x.Then, calculate du/dx:du/dx = d/dx (e2x) - d/dx (e-2x)du/dx = 2e2x - (-2e-2x) = 2e2x + 2e-2x = 2(e2x + e-2x).So, du = 2(e2x + e-2x) dx, which means (e2x + e-2x) dx = (1/2) du.Substitute u and du into the integral:∫ (1/u) ∙ (1/2) du = (1/2) ∫ (1/u) du= (1/2) ln|u| + C.Substitute back u = e2x - e-2x:= (1/2) ln|e2x - e-2x| + C. This matches option C.