Articles of calculus

Trig integral $\int{ \cos{x} + \sin{x}\cos{x} dx }$

Assume we have: $ \int{ \cos{x} + \sin{x}\cos{x} dx } $ 2 ways to do it: Use $\sin{x}\cos{x} = \frac{ \sin{2x} }{2} $ Then $ \int{ \cos{x} + \frac{\sin{2x}}{2} dx } $ $ = \sin{x} – \frac{ cos{2x} }{ 4 } + C $ Or the other way, just see that $ u = \sin(x), […]

Prove that all values satisfy this expression

I’m trying to find for what values for $p$ will cause $$\displaystyle\lim_{n\to\infty} \frac{\ln^p{n}}{n} = 0$$ I believe that one criteria for a limit approaching zero is that the top should be going to infinity slower than the bottom; however in this expression, it seems that I can make the top grows as fast as I […]

Prove $\int_{0}^{\pi/2} x\csc^2(x)\arctan \left(\alpha \tan x\right)\, dx = \frac{\pi}{2}\left$

When I showed to my brother how I proved \begin{equation} \int_{0}^{\!\Large \frac{\pi}{2}} \ln \left(x^{2} + \ln^2\cos x\right) \, \mathrm{d}x=\pi\ln\ln2 \end{equation} using the following theorem by Mr. Olivier Oloa \begin{equation}{\large\int_{0}^{\!\Large \frac{\pi}{2}}} \frac{\cos \left(\! s \arctan \left(-\frac{x}{\ln \cos x}\right)\right)}{(x^2+\ln^2\! \cos x)^{\Large\frac{s}{2}}}\, \mathrm{d}x = \frac{\pi}{2}\frac{1}{\ln^{\Large s}\!2}\qquad,\;\text{for }-1<s<1.\end{equation} He showed me the following interesting formula \begin{equation} \int_{0}^{\!\Large \frac{\pi}{2}} x\csc^2(x)\arctan […]

Show that $\int_0^\pi \log^2\left(\tan\frac{ x}{4}\right)dx=\frac{\pi^3}{4}.$

Hi I am trying to prove the relation $$ I:=\int_0^\pi \log^2\left(\tan\frac{ x}{4}\right)dx=\frac{\pi^3}{4}. $$ I tried expanding the log argument by using $\sin x/ \cos x=\tan x,$ and than used $\log(a/b)=\log a-\log b$, I get $$ I=\int_0^\pi \left( \log \sin \frac{x}{4}-\log\cos \frac{x}{4}\right)^2dx. $$ We can distribute this out $$ \int_0^\pi \log^2 \sin \frac{x}{4}dx +\int_0^\pi \log^2\cos \frac{x}{4}dx-2\int_0^\pi\log […]

Show $\sum_{k=1}^{\infty}\left(\frac{1+\sin(k)}{2}\right)^k$ diverges

Show$$\sum_{k=1}^{\infty}\left(\frac{1+\sin(k)}{2}\right)^k$$diverges. Just going down the list, the following tests don’t work (or I failed at using them correctly) because: $\lim \limits_{k\to\infty}a_k\neq0$ — The limit is hard to evaluate. $\lim \limits_{k\to\infty}\left|\dfrac{a_{k+1}}{a_k}\right|>1$—Limit does not converge; inconclusive. $\lim \limits_{k\to\infty}\sqrt[k]{|a_k|}>1$—Limit does not converge; inconclusive. $\int \limits_{1}^{\infty}a_k\,dk$—How do I even do this. There exists a $|b_k|\geq|a_k|$ and $\sum b_k$ converges—Cannot […]

how to compute the de Rham cohomology of the punctured plane just by Calculus?

I have a classmate learning algebra.He ask me how to compute the de Rham cohomology of the punctured plane $\mathbb{R}^2\setminus\{0\}$ by an elementary way,without homotopy type,without Mayer-Vietoris,just by Calculus. I have tried and failed.Is it possible to compute the de Rham cohomology just by Calculus?

Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?

In the book Thomas’s Calculus (11th edition) it is mentioned (Section 3.8 pg 225) that the derivative $\frac{\textrm{d}y}{\textrm{d}x}$ is not a ratio. Couldn’t it be interpreted as a ratio, because according to the formula $\textrm{d}y = f'(x)\textrm{d}x$ we are able to plug in values for $\textrm{d}x$ and calculate a $\textrm{d}y$ (differential). Then if we rearrange […]

Lesser-known integration tricks

I am currently studying for the GRE math subject test, which heavily tests calculus. I’ve reviewed most of the basic calculus techniques (integration by parts, trig substitutions, etc.) I am now looking for a list or reference for some lesser-known tricks or clever substitutions that are useful in integration. For example, I learned of this […]

Taylor's Theorem with Peano's Form of Remainder

The following form of Taylor’s Theorem with minimal hypotheses is not widely popular and goes by the name of Taylor’s Theorem with Peano’s Form of Remainder: Taylor’s Theorem with Peano’s Form of Remainder: If $f$ is a function such that its $n^{\text{th}}$ derivative at $a$ (i.e. $f^{(n)}(a)$) exists then $$f(a + h) = f(a) + […]

How to prove that $\lim \frac{1}{n} \sqrt{(n+1)(n+2)… 2n} = \frac{4}{e}$

I’d like a hint to show that: $$\lim \frac{1}{n} \sqrt[n]{(n+1)(n+2) \cdots 2n} = \frac{4}{e} .$$ Thanks.