the original Latent Dirichlet Allocation Algorithm ( about expanding expectation )

Each of the terms in equation 14 expands to one line in equation 15. To compute these expectations you just write down the expansions of the formulas for the probability densities needed and use regular algebraic identities. For example, for the first case, E[log P(theta|alpha)], note that the density of theta is a Dirichlet, so it becomes

E[log ((prod_i=1^k Gama(alpha_k))/Gamma(sum_i=1^k alpha_k)) 
          prod_i=1^k theta_k^(alpha_k-1 ))]

Expanding the logarithms you get

E[sum_i=1^k log Gama(alpha_k)  - log Gamma(sum_i=1^k alpha_k)  
    + sum_i=1^k  (alpha_k-1 ) log theta_k]

Using linearity of expectation then you get the first two terms in line 15. To get the last term, look up on wikipedia for the Dirichlet distribution the expectation of log theta_i, which is Digamma(alpha_i) - Digamma(sum_k alpha_k). Using the fact that the variational parameters for theta are named gamma then you have the final expression. For the other four lines in eq (15) you use essentially the same logic.