Write down:
#=int1/(x(x+3))dx#
#=int( { }/x + { }/(x+3))dx#.
Then apply the cover-up rule for partial fractions. To find out what goes over the #x#, use your finger to cover up the factor #x# in the denominator of the fraction on the first line, and replace all other #x#'s with zero. Similarly, to find out what goes over the #x+3#, cover up the #x+3# and replace the other #x# with #-3#. In each case, you replace #x# with whatever value of #x# makes the expression under your finger zero.
#=int( (1)/(0+3))/x + (1/(-3))/(x+3)dx#
#=1/3int1/x-1/(x+3)dx#.
