Question Detail

This integral is of the form ∫ f'(x)/f(x) dx = ln|f(x)| + C.Let the denominator be f(x) = e2x - e-2x.Then, calculate its derivative: f'(x) = d/dx(e2x - e-2x) = 2e2x - (-2)e-2x = 2e2x + 2e-2x = 2(e2x + e-2x).Notice that the numerator of the integrand is (e2x + e-2x), which is exactly (1/2)f'(x).So, we can rewrite the integral as:∫ (1/2) · (2(e2x + e-2x)) / (e2x - e-2x) dx = (1/2) ∫ f'(x)/f(x) dx= (1/2) ln|f(x)| + CSubstitute f(x) back:= (1/2) ln|e2x - e-2x| + C.