Deriving equations for memristor
I think that you have misunderstood the proposition of Chua: "Observe that the value of the incremental memristance (memductance) at any time \$t_0\$ depends upon the time integral of the memristor current (voltage) from \$t = - \infty \$ to \$t= t_0\$."

In fact the observation just says that \$M \$ is function of the integral of the current 
$$
M=M (q(t_0))=M\left ( \int_{-\infty}^{t_0}i (\tau)\mathrm d \tau\right)
$$
and that $W $ is function of the integral of the voltage 
$$
W=W (\varphi(t_0))=W\left ( \int_{-\infty}^{t_0} v(\tau)\mathrm d \tau\right)
$$

You have nothing to calculate.