Have you determined the partial fraction decomposition? Once you do, then the integration is straightforward. Are you familiar with the cover-up method?
That's the correct partial fraction decomposition. In the second line, instead of an integral sign, you want -ln(x+1)+ln(x+2) evaluated from 0 to 1, so it's -ln(1+1)+ln(1+2)+ln(0+1)-ln(0+2). Now you can combine the logarithms to get k.