A calculus problem by Rohith M.Athreya

Felt like a level 5 to me.

First let us state a small Lemma:-

If $f(x)$ is a polynomial function then $\large \int_{a}^{b} e^x . f(x) \, dx$ is given by $e^x.(f(x)-f'(x)+f''(x) ... \pm f^{(n)}(x)) +c$ where n is the degree of given polynomial.

It can be proved by differentiating the result and all terms cancel to give our initial polynomial back.

So now we can tackle the question.

$\large \int_{e}^{e^{2}} (\ln x)^{81} \, dx$

Let $x=e^t \text{ and } dx=e^t.dt$

$\large \int_{1}^{2} (t)^{81} .e^t\, dt$

We can now apply our lemma on this to get the answer as $[e^t.(t^{81}-81.t^{80}+81.80.t^{79} ... -81!)]^2_1$ This is where I keep my pen and paper down and write a program to do the remaining calculations. It returns the answer as 4.26385564883e+23 and the ratio as 0.707893621242