Question Detail

To evaluate the integral ∫ (e2x + e-2x) / (e2x - e-2x) dx, use u-substitution.Let u = e2x - e-2x.Then, find du/dx:du/dx = d/dx(e2x) - d/dx(e-2x)= (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.The integral of (1/u) with respect to u is ln|u|.= (1/2) ln|u| + C.Finally, substitute back u = e2x - e-2x:= (1/2) ln|e2x - e-2x| + C.