Let I(n) = integral from 1 to e of (ln(x)^n)/(x^2) dx where n is a natural number. Firstly find I(0). Show that I(n) = -(1/e) + n*I(n-1). Using this formula find I(1).

For the first part this is a standard polynomial integral whereby we can follow the power rule for anti-differentiation. In part ii), if we look at the question we see that we're trying to get I(n-1) in our formula for I(n). Now instantly you should recognise that using a function, in this case ln(x), to the power of n when differentiated gives, by the chain rule, that function to the power of n-1 multiplied by n and the derivative of ln(x). Since we have nI(n-1), it is clear this is the route we should go down so now we consider how we can rewrite I(n) in terms of the derivative of our function. Integration by parts is clearly the method to use as it involves differentiation and since we want to differentiate (ln(x))^n let u(x) = ln(x)^n, and v'(x) = x^(-2) so we have u'(x) = (-n/x)(ln(x))^(n-1) and v(x) = -1/x. Once we plug everything in and rearrange, computing the limits for the u(x) * v(x) yields the result easily. For the last part simply plug in I(0) and we get 1 - 2/e. The crucial part of this question was seeing that n*I(n-1) looks a lot like the derivative of I(n) since we have a function to the power of n.