0
votes
0answers
39 views

Integral of a function and its inverse

This comes from the comments section of this question. The original question was to show the following identity for some increasing invertible function $f$ ...
4
votes
2answers
170 views

Evaluate $\sum_0^\infty \frac{1}{n^n}$

Courtesy of this xkcd comic I now know that $$ \sum_{n=1}^\infty \frac{1}{n^n} \approx \ln^e(3) $$ Echoing the views of the comic itself, if I ever find myself taking $\ln^e(x)$ then something has ...
5
votes
1answer
76 views

Definite Integral that Evaluates to Teacher's Initials: TAA

My school's calculus teacher's birthday is in a couple of days, and our class decided to give him a surprise birthday card that has a definite integral which evaluates to his initials (TAA). So far ...
5
votes
2answers
174 views

Compute the integral $\int_{0}^{2\pi}|\cos^n t|\ dt $ for $n \in \mathbb{Z}$

I need some help on computing this integral. I thought of this when solving another question on S.E. Evaluate$$I=\int_{0}^{2\pi} |\cos^n t|\ dt $$ for $n \in \mathbb{Z}$. Observation: This ...
2
votes
2answers
37 views

Do I have this right Trip integrals

Just finished my proof of the volume of a cone using trip integrals. I think I noticed something. Wonder if I got it right. The first integral defines the line/curve, the second defines the area ...
7
votes
1answer
209 views

Evaluation of the integral $\int_0^1 \log{\Gamma(x+1)}\mathrm dx$

As it says in the title, I'd like to know how to solve the definite integral $\int_0^1 \log{\Gamma(x+1)}\mathrm dx$. Mathematica gives the answer $\frac{1}{2}\log (2\pi)-1$ but I have no idea how one ...
-3
votes
1answer
45 views

Computation of integral [closed]

I want to compute this integral: \begin{equation*} J=\int_{0}^{1}\ln(p)\ln(1-p)p^{2}dp \end{equation*} It will be great if you can detail the proof. I tryed to do change of variable it does not ...
83
votes
5answers
4k views

Help find hard integrals that evaluate to $59$?

My father and I, on birthday cards, give mathematical equations for each others new age. This year, my father will be turning $59$. I want to try and make a definite integral that equals $59$. So ...