So I was looking through the homepage of Youtube when I came across this video by Blackpenredpen that was of him solving a joke improper integral that was supposed to represent how much time he spent on Youtube-related tasks. Here is the improper integral:$$\left(\int_{\sum_{n=0}^\infty(\sqrt n-\sqrt{n+1})}^{\sum_{n=0}^\infty\frac{(-1)^n}{n+1}}\left(\lim_{t\to\infty}\left(1+\frac{1}{e^t}\right)^{e^t}\right)^{\frac{d}{dx}\left(\frac{x^2}{\sin^2(x)+\cos^2(x)}\right)}dx\right)^2$$ $\small\text{I mean at least I think it was a joke}$
Anyways, here is my attempt at evaluating the improper integral:
Right away, we can solve the upper limit of the integral, since I remembered from past experience that the sum for it was equivalent to $\ln(2)$, and that the lower integral is equivalent to $-\infty$, so we can right away rewrite this as$$\left(\int_{-\infty}^{\ln(2)}\left(\lim_{t\to\infty}\left(1+\frac{1}{e^t}\right)^{e^t}\right)^{\frac{d}{dx}\left(\frac{x^2}{\sin^2(x)+\cos^2(x)}\right)}dx\right)^2$$Now, to solve the limit. We can let $s=e^t$ to rewrite the limit as $\lim_{s\to\infty}\left(1+\frac{1}{s}\right)^s$, which equals $e$. And now, to solve the derivative, since I know that $\sin^2(x)+\cos^2(x)=1$, we can rewrite it as $\dfrac{d}{dx}x^2=2x$, which now we can rewrite the equation as$$\left(\int_{-\infty}^{\ln(2)}e^{2x}dx\right)^2$$which now we can rewrite as$$\left(\frac{1}{2}\left.e^{2x}\right]_{-\infty}^{\ln(2)}\right)^2$$$$=\left(\left(\frac{1}{2}e^{2\ln(2)}\right)-\left(\frac{1}{2}e^{2(-\infty)}\right)\right)^2$$$$=\left(\left(\frac{1}{2}\cdot4\right)-\left(\frac{1}{2}\cdot0\right)\right)^2$$$$=4-0=4\text{ hours a day for Youtube-related activities}$$
$\small\text{Which if I'm being honest, that's pretty much }\frac{1}{3}\small\text{rd to }\frac{1}{4}\small\text{th the amount of time that I'll spend on Youtube a day usually.}$
My question
Is my evaluation correct, or what could I do to attain the correct solution/attain it more easily?
Mistakes I might have made
- Simplifying the summation
- Trigonometric identities
- The derivative
- The integral
- Calculation errors