heaps the curve of the log the area under the curve can be found by taking the definite integral from to nwhich in the picture is but in general would be from this we get the following inequality  log dx < log = nowconsider shifting the entire green area to the right by one in the figure abovethat' the orange area the orange and green areas are exactly the same size the orange is just shifted right by one now look at the plot of the log base of the area below the curve is now clearly bigger than the orange area if we imagine this graph going out to nthen we'll have to include in our definite integral (since we shifted the orange area to the rightso we get the following inequality + log <log dx = putting the two inequalities together we have lower and upper bound for our summation +  log dx <log <log dx = it is easier to integrate using natural log so we'll rewrite the integral as follows   ln dx log dx ln the constant term in the integral can be factored out so we'll look at the following integral  ln dx we can find the result of the definite integral that appears above by doing integration by parts the integration by parts rule is as follows   dv uvv du this copy belongs to 'acha