Product of cdf and pdf of normal distribution. 
A continuos random variable $X$ has the density
  $$
f(x) = 2\phi(x)\Phi(x), ~x\in\mathbb{R}
$$
  then
(A) $E(X) > 0$
(B) $E(X) < 0$
(C) $P(X\leq 0) > 0.5$
(D) $P(X\ge0) < 0.25$
\begin{eqnarray}
\Phi(x) &=& \text{Cumulative distribution function of } N(0,1)\\
\phi(x) &=& \text{Density function of } N(0, 1)
\end{eqnarray}

I don't have a slightest clue where to start with. Can someone give me a little push. I saw some answers on same question like this but I didn't understand how should I integrate it when calculating expectation. 

Yes, you expect the higher of two samples to be greater on average than what you expect for just the first sample, since the second sample can only make the maximum go up.  (Remember $0$ is the expected value of a single sample of $N(0,1)$.)
B: No, you don't expect it to be lower.
C: Will the greater of two samples of $N(0,1)$ be negative more than half the time?  Of course not!  (It will be negative 1/4 of the time -- when both samples are negative.)
D: Will the greater sample be positive less than 1/4 of the time?  Of course not!  (It will be positive 3/4 of the time -- when either sample is positive.)
