Question Detail

This integral can be solved using a simple substitution.Let u be the denominator: u = e2x - e-2x.Now, find the derivative of u with respect to x (du/dx):du/dx = d/dx (e2x) - d/dx (e-2x)du/dx = (2e2x) - (-2e-2x)du/dx = 2e2x + 2e-2x = 2(e2x + e-2x)From this, we get du = 2(e2x + e-2x) dx.Rearranging for (e2x + e-2x) dx, we have (e2x + e-2x) dx = du/2.Substitute u and du into the integral:∫ (1/u) * (du/2) = (1/2) ∫ (1/u) duIntegrating 1/u gives ln|u|:= (1/2) ln|u| + CFinally, substitute u = e2x - e-2x back:= (1/2) ln|e2x - e-2x| + C.This matches option C.