(Core 2) Show that the region bounded by the curve y = 7x+ 6 - (1/x^2), the x axis and the lines x = 1 and x = 2 equals 16

The question wants us to confirm the answer so the marks will be in the method unlike most questions which require you to find the answer. How to approach this is by thinking what ways can we calculate areas from equations. We could split the graph up into shapes is we knew what the graph looked like but as it is quite complicated it only leaves us with one method which is integration. Recall that to intergrate an equation we need to increase the power (indice) by 1 and then divide the coefficient (Number in front of the variable) by the new power. Let’s apply this to the question...
as x is just x¹. 7x becomes (7/2)x²likewise with 6 as it’s a constant which is 6x⁰ it becomes just 6xnow the last term can be a bit confusing with its current form so if we convert it into a more friendly format we can reduce mistakes and more importantly marks! Recall 1/x = x^-1so we can write - (1/x^2) as -x^-2 now being carefully with negatives we can do the integration which goes to x^-1
which leaves us with (7/2)x² + 6x + x^-1 BUT remember +c!However as we are given limits to this integral (Fancy name of an expression we’ve intergrated) we don’t need the plus c and just substitute the limits in to find our answer
(7/2)2² + 6(2) + 2^-1 - ( (7/2)1² + 6(1) + 1^-1) = 14 + 12 + 1/2 - ( 7/2 + 6 + 1) = 16 as required (substituted upper limit minus substituted lower limit)