For basic questions about limits, derivatives, integrals, and applications, mainly of one-variable functions.

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0
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1answer
13 views

2 question about supremum of subset and a sequence that converge to it.

Let $A$ be a bounded subset of $\mathbb{R}$. 1. Show that there exists a sequence $a_n$ of elements of $A$ such that $\lim _{ }\left(a_n\right)\:=\:sup\left(A\right)$ 2. Show that we can build a ...
3
votes
0answers
42 views

Evaluation of $\int \frac{\sqrt{1+x^4}}{1-x^4}dx$

Evaluation of $\displaystyle \int \frac{\sqrt{1+x^4}}{1-x^4}dx$ $\bf{My\; Try::}$ Given $\displaystyle \int\frac{\sqrt{1+x^4}}{1-x^4}dx\;,$ Then We can write the above Integral as $$\displaystyle ...
2
votes
1answer
61 views

How to evaluate this indefinite integral $\int\frac{\cos(x)}{1+\mathrm{e}^x}\mathrm{d}x$ [duplicate]

One of my student asked me to help her evaluate this indefinite integral $$\int\dfrac{\cos x}{1+e^x}\mathrm{d}x,$$ and I tried several minutes, but at last I had to given up, for I thought that it is ...
3
votes
2answers
33 views

Quadratic Expressions: Advanced techniques of Integration

$\int$ $\frac{x}{\sqrt{5+12x-9x^2}}\,dx$ After two steps I arrive at $\int \frac{x}{\sqrt{9-(3x-2)^2}}\,dx$ Using trigonometric substitution, we have a triangle with a cosine of $\theta$ of ...
0
votes
0answers
26 views

Advanced Integration techniques: Quadratic Expressions and U-Substitution

Find $\int$ $ dx(2x-1)\over (x^2-6x+13) $ In the final steps after a u-substitution, one arrives at $\int$ $dx(2u)\over(u^2+4)$ + $\int$ $(5du)\over (u^2+4)$ The next step is arriving at ln(u^2+4) + ...
1
vote
1answer
20 views

Finding pathline

I've been trying to find the pathline of a particle dropped in a steady flow defined by the following vector components: $$ u= \frac{-2x}{(x^2+y^2+1)^2} \hat i + \frac{-2y}{(x^2+y^2+1)^2}\hat j $$ in ...
1
vote
2answers
43 views

Solutions to the integral $\int \frac {dx}{2\sqrt x (x+1)}$

I am given a question to solve the integral $\int \frac {dx}{2\sqrt x (x+1)}$. When I substitute $x+1 = t^2$, I get the solution as $\space \ln(\sqrt{x+1} + \sqrt x) +C$; while when I substitute ...
1
vote
3answers
49 views

A quadratic polynomial is nonnegative for all $x$ if and only if the discriminant is nonpositive

Show that if $a>0$ the inequality $ax^2+2bx+c\ge 0 $ for all values of $x$ if and only if $b^2-ac\le 0$. I tried to prove it by: $ax^2+2bx+c≥ b^2-ac$. Used partial derivatives with respect to ...
0
votes
1answer
48 views

differential inequality implies zero function

Let $f$ be twice continuously differentiable on $(-1,1)$, and $f(0)=f'(0)=0$, $\quad|f''(x)|\leq |f(x)|+|f'(x)|$. Show that $f=0$ in some neighborhood of $0$. How can we deduce something from this ...
0
votes
2answers
25 views

Evaluation of an integral of some expressions involving fractions

I am stuck in evaluating the following integral: \begin{equation} \int_{0}^{b-a} \frac{1}{\sqrt{u} (a+u)} \,du, \end{equation} where $0<a<b$. Any ideas?
1
vote
1answer
23 views

Mean value inequality geometrical interpretaion

The mean value inequality theorem Let U be an open interval in $\mathbb{R}$. Suppose that $K \ge 0$ and that, $a,b \in U$ with $b>a$. If $f : U \rightarrow \mathbb{R}$ is differentiable with ...
0
votes
0answers
15 views

Additive & Multiplicative Link

Is there an isolative property about the following multivariable equation: $$f_1(x)g_1(y)=f_2(x)+g_2(y)$$ That is, is this equation rearrangable such that it may be put in the form ...
5
votes
2answers
28 views

Integration of high order fraction; explanation of method.

I understand the cases where the order is less or equal to 3 (example where it is three we split numerator with A,B,C), but in this case(Example 8) I do not see why we split the numerator with A, ...
1
vote
1answer
18 views

$k_{n+1}\le (1+2\varepsilon)k_n$ for $k_n:=\lfloor(1+\varepsilon)^n\rfloor$ and $\varepsilon>0$

Let $$k_n:=\lfloor(1+\varepsilon)^n\rfloor\stackrel{\text{def}}{=}\max\left\{k\in\mathbb{Z}:k\le(1+\varepsilon)^n\right\}\;\;\;\text{for }n\in\mathbb{N}$$ How can we prove $k_{n+1}\le ...
0
votes
0answers
17 views

Fourier series - Understanding an equality

Why is this equality true: $$\left\langle {f,g} \right\rangle = \sum\limits_{n = - N}^N {\hat{f}(n)\hat{g}(n)}$$ where $$f = \sum_{n=-N}^N c_n e^{int}, g=\sum_{n=-N}^N d_n e^{int} $$ and ...
0
votes
0answers
27 views

Using the Maclaurin series to approximate $f(0.1)$ for $f(x)=(3+e^{2x})^{0.5}$

I was tasked to use the Maclaurin series to calculate $f(0.1)$ of $f(x)=(3+e^{2x})^{0.5}$. I got the Maclaurin expansion of $p_2(x) = \sqrt{3} + 4x +5x^2$ into which I plugged $0.1$ to yield ...
1
vote
0answers
33 views

What's wrong with my reasoning while setting up a limit?

I was writing an answer to this question, which asks about what happens to the apex of an isosceles triangle if a vertex is at infinity. I thought it would be very easy to prove it by setting up a ...
0
votes
2answers
35 views

Proof for pythagoras theorem

Let $f,g$ orthogonals to each other. $${\left\| {f + g} \right\|^2} = \left<f,f\right>+\left<g,f\right>+\left<f,g\right>+\left<g,g\right> = {\left\| f \right\|^2} + {\left\| g ...
1
vote
0answers
8 views

A treatise on Probabilistic arguments (or even Laplace/Fourier transforms) to solve limits/integrals from basic calculus.

I've seen in some answers in Brilliant.org to some very complicated limits and integrals that uses probabilistic arguments (Let $X$ be a random variable from $[0,1]$... some examples are in those ...
5
votes
4answers
148 views

Finding $ \int_0^1 \frac {\ln x}{1+x^2}\mathrm dx $

Today I encountered the problem of how to find $$ \displaystyle\int_{0}^{1} \frac {\ln x}{1 + x^2}\mathrm dx $$ but got no start on it. Is this one of those integrals which we have to approach from ...
3
votes
2answers
81 views

Probabilistic techniques, methods, and ideas in (“undergraduate”) real analysis

As the book Probabilistic Techniques in Analysis by Richard Bass shows, nowadays techniques drawn from probability are used to tackle problems in analysis. The mentioned book presents a survey of ...
2
votes
4answers
53 views

Prove that a function is continuous for every $x \in R$

Prove that the function: $$ f(x)=\frac{\sqrt{x^2-x+1}}{|\sin(x)-4|-2} $$ is defined for every $x \in R$ and continuous in every $x \in R$, So I said that in order for this function to be defined we ...
3
votes
3answers
58 views

Find $ \int \frac {1-x^2}{1+3x^2+x^4} \, \mathrm{d}x $

Today, the CalcBee sample problems got released. The following problem was my creation and I wanted to see how many solutions people can come up with. The result is very beautiful and I thought it ...
5
votes
4answers
114 views

Evaluating $\int{\frac{1}{\sqrt{x^2-1}(x^2+1)}dx}$

Evaluating $$\int{\frac{1}{\sqrt{x^2-1}(x^2+1)}dx}$$ using $ux=\sqrt{x^2-1}$ I try to $u^2x^2=x^2-1$ $x^2=\frac{-1}{u^2-1}$ However I cant get rid of $x$ because derivative has $x\;dx$. How can I ...
4
votes
1answer
24 views

Trigonometic Substitution VS Hyperbolic substitution

The following tables were taken from University of Pennsylvania's page about Calculus: Trigonometric Substitution Hyperbolic Substitution As you can see, the forms $1+x^2$ and $x^2-1$ are repeated ...
1
vote
4answers
37 views

Calculate a limit of exponential function

Calculate this limit: $$ \lim_{x \to \infty } = \left(\frac{1}{5} + \frac{1}{5x}\right)^{\frac{x}{5}} $$ I did this: $$ ...
5
votes
2answers
85 views

Closed form of $\int_0^1\int_0^1\int_0^1\frac{\left(1-x^y\right)\left(1-x^z\right)\ln x}{(1-x)^3}\,\mathrm dx\;\mathrm dy\;\mathrm dz$

While trying to find several references to answer Pranav's problem, I encounter the following multiple integrals $$I=\int_0^1\int_0^1\int_0^1\frac{\left(1-x^y\right)\left(1-x^z\right)\ln ...
0
votes
1answer
35 views

sequence defined by norm

Let $(u_n)_n$ be defined by: $\quad \begin{cases}u_1=1 & \\ \\ u_n=\left( \sum\limits_{k=1}^{n}u_{k}\right)^{\frac{1}{2}} & \end{cases} $ Show that $u_{n}\to +\infty$ and $u_n \simeq ...
1
vote
0answers
83 views

Evaluation of $\displaystyle\int\frac{1}{x^4+1}$ (Spivak's Calculus, Chapter 19, Problem 6viii) [duplicate]

Solve the integral:$$\int \frac{1}{x^4+1}\mathrm{d}x$$ I tried the substitution $x=\tan\theta \Rightarrow dx=\sec^2 \theta\,\mathrm{d}\theta$, but that leads to a dead end. $$\begin{align}\int ...
1
vote
1answer
17 views

Check my answer - simple laplace transform of piecewise continuous function.

I'd just like to check that I got the idea right, first exercise im doing in laplace transforms and am a bit clueless. We are given $f(t)=0$ if $0<t<2$ and $f(t)=t$ if $t>2$. We are asked to ...
2
votes
3answers
60 views

How can you evaluate this limit? [on hold]

How can you calculate this limit? $$ \lim_{x \to 1} \frac{e^{x-1}-1}{x^2-1} $$ I really don't have a clue what to do with the $e^{x-1}$
4
votes
2answers
149 views

Finding $\lim_{x\to 0}\frac{\sin(x+x^3/6)-x}{x^5}$

I'm trying to find the limit of this expression: $$\lim_{x\to0}\frac{\sin\left(x+x^3/6\right)-x}{x^5}$$ My solution is as follows: $$ \begin{align} ...
0
votes
7answers
116 views

Prove that $\lim_{x \to 0 } \frac{\ln(x+1)}{x} = 1$

I've looked around to see a proof for this limit and encountered this: $$ \lim_{x \to 0 } \frac{\ln(x+1)}{x} $$ $$ \lim_{x \to 0 } \frac{1}{x} \ln(x+1) $$ $$ \lim_{x \to 0 } \ln(x+1)^\frac{1}{x} ...
6
votes
2answers
134 views

Proof of expression with integrals

I have had trouble proving the following expression. Do you have any hints to help me? Let $f:[a,b]$ be an integrable function for which $$\int_a^bf(x)dx=6$$ Prove that there exist $t_1,t_2\in(a,b)$ ...
3
votes
3answers
66 views

Multiple choice question about limits and continuity? (Or, $\tan x$ is continuous?!)

I'm doing a test about limits and continuity and got these two wrong. $\mathbf{Q1}$: The function $f(x) = \tan x$: $\hspace{1em}\mathtt{a)}$ is continuous $\hspace{1em}\mathtt{b)}$ is ...
1
vote
2answers
30 views

Generalization of linear approximation? [on hold]

How is the linear approximation is generalized to the Taylor series? I do not get that concept.
5
votes
0answers
55 views

Density of the rationals in the reals

While studying measure theory I have encountered the following set, $$U_\varepsilon=\bigcup_{n\in \mathbb{N}}(q_n-\varepsilon /2^n,q_n+\varepsilon/2^n),$$ where $(q_n)_{n\in \mathbb{N}}$ is an ...
6
votes
2answers
232 views

Proving a limit using another limit

Let $f(x)$ be a functions that's defined at some neighbourhood of $0$ $$\lim\limits_{x \to 0} \frac{f(x)}{x} = 3$$ Prove that: $$ \lim_{x \to 0}\frac{f(3x)}{\ln(1+4x)} = 2.25 $$ I really ...
1
vote
1answer
16 views

Mass of the body M, Cartesian reference frame.

Oxyz is a Cartesian frame of reference with unit base vectors, $i,j$ and $k$. A rigid body $V$, of uniform density $p$, is bounded by the surfaces $y=(1-x^2)^{(1/2)}, z=0, y=0$ and $z=1-y$ If the mass ...
1
vote
1answer
46 views

Evaluate this infinite product involving $a_k$

Let $a_0 = 5/2$ and $a_k = a_{k-1}^{2} - 2$ for $k \ge 1$ Compute: $$\prod_{k=0}^{\infty} 1 - \frac{1}{a_k}$$ Off the bat, we can seperate $a_0$ $$= -3/2 \cdot \prod_{k=1}^{\infty} 1 - ...
3
votes
4answers
116 views

If we know $x+y+z=1$, $x^2+y^2+z^2=2$, and $x^3+y^3+z^3=3$, how to find $x^4+y^4+z^4$?

Let $x$, $y$, and $z$ be such that $$\begin{align*} x+y+z&=1\\ x^2+y^2+z^2&=2 \\ x^3+y^3+z^3&=3 \end{align*}$$ Then $x^4+y^4+z^4=?$
4
votes
0answers
46 views

Seperating single integral into an double integral.

Please refer to : How to prove that $\int_{0}^{\infty}\sin{x}\arctan{\frac{1}{x}}\,\mathrm dx=\frac{\pi }{2} \big(\frac{e-1}e\big)$ The answer by @Venus. What is the procedure in converting that ...
0
votes
3answers
54 views

Why don't graphing tools represent holes in a graph?

Why don't graphing tools represent holes in the graph of a function? A hole at a point in a graph is point where function is not defined. Suppose there is a function $$\frac{x}{\sqrt{x+1}-1}$$ Its ...
1
vote
0answers
15 views

Show that weak local minimum of a convex function $\mathbb{R}^N\rightarrow \mathbb{R}$ is its weak global minimum.

Show that weak local minimum of a convex function $\mathbb{R}^N\rightarrow \mathbb{R}$ is its weak global minimum. Does the same happen to strong minimums? I know that when $f$ is convex, then we ...
-3
votes
0answers
29 views

How to find the area within the curve $ a^2 \cdot y^2 = x^3(2a - v) $? [on hold]

Here is the equation of a curve: $ a^2 \cdot y^2 = x^3(2a - v) $. Now I want to find the whole area of the curve . How can I find the whole area?
5
votes
1answer
74 views

Multiple integrals involving product of gamma functions

The following integral was posted a few days back on Integrals and Series forum: $$\int_0^{2\pi} \int_0^{2\pi} \int_0^{2\pi} \frac{dk_1\,dk_2\,dk_3}{1-\frac{1}{3}\left(\cos k_1+\cos k_2+ \cos ...
1
vote
0answers
15 views

Shifting integration variables

I'm not sure how to pose this question precisely, but I'll try. I'm trying to see what happens when you have an integral of the form $\int \mathrm{d}x \,f(x-g(z))$ and you try and write it as $\int ...
3
votes
2answers
75 views

Surface area of a solid of revolution: Why does not $ \int_{b}^{a} 2\pi \,f(x) \,dx $ work?

Why does not $ \int_{b}^{a} 2\pi \,f(x) \,dx $ yield the correct answer when calculating the surface area of a solid of revolution?
1
vote
2answers
67 views

Why does this inequality stand?

I want to ask something about: "Since $i \log_e i$ is concave upwards, it is easy to show that $$\sum_{i=2}^{n-1} i \log_e i \leq \int_2^n x \log_e x \,dx \leq \frac{n^2 \log_e ...
2
votes
1answer
42 views

Determining a radius convergence of a power series

Let $$ \sum_{n=0}^\infty \frac{(-1)^n}{3n+1} x^{3n+1} $$ Is there an immediate way to determine $R=1$?