The attached file contains part of page 151 from D'Inverno's Introducing Einstein's Relativity. Beneath the shaded equation (11.38) he continues with 'Integrating the first two terms in (11.37) by parts, we get....' ....equation (11.39). This doesn't look right to me because the original equation has one term on the left hand side and three terms on the right hand side but in the resultant equation, two of the terms (the LG on the LHS and the LG barred on the RHS) have not been integrated. How can you get away with integrating only part of an equation??