Question Detail

This integral is of the form ∫ f'(x)⁄f(x) dx = ln|f(x)| + C.Let u = e2x - e-2x (the denominator).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.This means (1⁄2) du = (e2x + e-2x) dx (the numerator).Substitute u and (1⁄2)du into the integral:∫ 1⁄u (1⁄2) du = 1⁄2 ∫ 1⁄u du= 1⁄2 ln|u| + CSubstitute back u = e2x - e-2x:= 1⁄2 ln|e2x - e-2x| + C.This matches option C.