If the integral of $\frac{dQ}{T}$ over an engine cycle is less than 0 , then why is entropy of the universe always increasing? For a reversible process, we define $ dS= \frac{dQ}{T}$ , so, the integral being negative would suggest that the entropy of universe decreases with each cycle of the engine because the clasius inequality states this quantity is less than 0 
 A: The value of $\Delta S_{\text{sys}}$
You are confusing the entropy change of the system with the total entropy change. Now since the process is cyclic, the total entropy change for the system will be zero ($\Delta S_{\text{sys}}=0$). It does not matter whether the process is reversible or irreversible because entropy is a state function and depends only on the initial and the final states, which in this case are the same.
What about $\Delta S_{\text{surr}}$?
Now, I would like to state the Clausius inequality more clearly:
$$\oint\frac{\delta Q_{\text{sys}}}{T_{\text{surr}}}≤0\tag{1}$$
As you can see, the temperature in the denominator is the temperature of the surroundings, not the system, and $\delta Q_{\text{sys}}$ is an inexact differential of the heat given to the system. Since the amount of heat given to the system will be equal to the amount of heat lost from the surroundings, thus
$$\delta Q_{\text{sys}}=-\delta Q_{\text{surr}}\tag{2}$$
So, now let's compute $\Delta S_{\text{surr}}$ (entropy change of the surroundings) for a cyclic process:
$$\Delta S_{\text{surr}}=\oint \frac{\delta Q_{\text{surr}}}{T_{\text{surr}}}\tag{3}$$
But using equation $(2)$, we can rewrite the above equation as:
$$\Delta S_{\text{surr}}=-\oint \frac{\delta Q_{\text{sys}}}{T_{\text{surr}}}\tag{4}$$
Now, using equation $(1)$, we can conclude that
$$\Delta S_{\text{surr}}=-\oint \frac{\delta Q_{\text{sys}}}{T_{\text{surr}}}≥0\Longrightarrow \Delta S_{\text{surr}}≥0\tag{5}$$
Total entropy change
The total entropy change is given by
\begin{align}
\Delta S_{\text{total}}&=\underbrace{\Delta S_{\text{sys}}}_{=0}+\underbrace{\Delta S_{\text{surr}}}_{≥0}\\
\therefore \Delta S_{\text{total}}&≥0\tag{6}
\end{align}
Thus, as you can see, Clausius inequality also yields the fact that the total entropy of the universe must increase. Do note that the equality in the equation $(6)$ holds when the process is reversible.
