Question Detail

This integral can be solved using substitution.Let u be the denominator: u = e2x - e-2x.Now, find the differential du by differentiating u with respect to x:du/dx = d/dx(e2x) - d/dx(e-2x)du/dx = (2e2x) - (-2e-2x)du/dx = 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 (1/2)du into the integral:∫ (1/u) * (1/2) du = (1/2) ∫ (1/u) duThe integral of (1/u) is ln|u|.= (1/2) ln|u| + CSubstitute back u = e2x - e-2x:= (1/2) ln|e2x - e-2x| + C.This matches option C.