Articles of calculus

Very confused about a limit.

This question is about where I made my mistake in the computation of a limit. It relates to An answer I gave that confused me. The question to which I gave the (partial) answer is related to tetration but my mistake is probably a simple general one ( considering tetration as complicated ). Here is […]

Evaluate $\displaystyle I=\int _{ 0 }^{ 1 }{ \ln\bigg(\frac { 1+x }{ 1-x } \bigg)\frac { dx }{ x\sqrt { 1-{ x }^{ 2 } } } }$

Problem: Evaluate: $$\displaystyle I=\int _{ 0 }^{ 1 }{ \ln\bigg(\frac { 1+x }{ 1-x } \bigg)\frac { dx }{ x\sqrt { 1-{ x }^{ 2 } } } }$$ On Lucian Sir’s advice, I substituted $x=\cos(\theta)$. Thus, the Integral becomes $$\int_0^{\pi/2} \ln\bigg(\dfrac{1+\cos(\theta)}{1-\cos(\theta)}\bigg)\dfrac{1}{\cos(\theta)}d\theta$$ $$=\int_0^{\pi/2} \ln\bigg(\cot^2\dfrac{\theta}{2}\bigg)\dfrac{1}{\cos(\theta)}d\theta$$ Unfortunately I’m stuck now. I would be indeed grateful if somebody […]

Does $1 + \frac{1}{x} + \frac{1}{x^2}$ have a global minimum, where $x \in \mathbb{R}$?

Does the function $$f(x) = 1 + \frac{1}{x} + \frac{1}{x^2},$$ where $x \in {\mathbb{R} \setminus \{0\}}$, have a global minimum? I tried asking WolframAlpha, but it appears to give an inconsistent result.

Derivative of a rational function

I’ve found the following derivative in my Calculus book and I can’t get my my head around the algebra involved. Can anybody help me? Thanks.

Please explain the logic behind $d(xy) = y(dx) + x(dy)$

I’ve seen $d(xy) = y(dx) + x(dy)$, but I don’t understand the principle behind it and memorizing it is lame. Can anyone explain what is going on here? For example from physics, $$F = {{dP} \over {dt}}$$ $$F = {{d(mv)} \over {dt}}$$ $$F = {{v(dm) + m(dv)} \over {dt}}$$ Since $$dm = 0$$ $$F = […]

Taylor's series when x goes to infinity

Let $f(x) = \frac {x^3}{(x+1)^2}$. Find constants a, b, c, so that $f(x) = ax + b + \frac cx + o(\frac 1x)$ as $x$ goes to $\pm \infty$. So i know that i can’t take Taylor series as $x$ goes to infinity. So i am assuming i have to make some kind of substitution. […]

Improper integral of $\int_0^\infty \frac{e^{-ax} – e^{-bx}}{x}\ dx$

This question already has an answer here: proving of Integral $\int_{0}^{\infty}\frac{e^{-bx}-e^{-ax}}{x}dx = \ln\left(\frac{a}{b}\right)$ 3 answers

What justifies writing the chain rule as $\frac{d}{dx}=\frac{dy}{dx}\frac{d}{dy}$ when there is no function for it to operate on?

This previous question of mine has lead me to ask the following question: It was my understanding that the chain rule $$\dfrac{du}{dx}=\dfrac{dy}{dx}\dfrac{du}{dy}$$ only makes sense when there is some function $u$ for it to operate on. So how can we possibly justify writing $$\dfrac{d}{dx}=\dfrac{dy}{dx}\dfrac{d}{dy}?$$ One of the answers to the previous question mentioned that if […]

Integral of $\csc(x)$

I’m getting a couple of different answers from different sources, so I’d like to verify something. I misplaced my original notes from my prof, but working from memory I have the following: \begin{align} \int\csc(x)\ dx&=\int\csc(x)\left(\frac{\csc(x)-\cot(x)}{\csc(x)-\cot(x)}\right)\ dx\\ &=\int\frac{\csc^{2}(x)-\csc(x)\cot(x)}{\csc(x)-\cot(x)}\ dx\\ &=\int\frac{1}{u}\ du\\ &=\ln|u|+C\\ &=\ln|\csc(x)-\cot(x)|+C \end{align} This looks proper when I trace it, but wolfram alpha is saying […]

Evaluate the sum $\sum^{\infty}_{n=1} \frac{n^2}{6^n}$

Evaluate the sum $\sum^{\infty}_{n=1} \frac{n^2}{6^n}$ My approach : $= \frac{1}{6}+\frac{2^2}{6^2}+\frac{3^2}{6^3} +\cdots \infty$ Now how to solve this I am not getting any clue on this please help thanks.