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

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0
votes
1answer
7 views

Evaluate the integral, and then take the derivative of it.

I'm mostly curious as to if the way I've went about solving this is correct, or if there is a more simple way to get the answer. So I first evaluated the top section And when I did that I got ...
-1
votes
1answer
19 views

Y=0 plane picture

Is the y=0 plane the combination of the xy and zy plane because x and z are not restricted, or is it just the yz plane. Wolfram Alpha shows it just as the yz plane.
0
votes
1answer
40 views

How to rewrite a derivative w.r.t. tensor as w.r.t. vector

I'm stuck on a (probably very simple) problem I've come across. Take a function $f(A)$ where $A$ is a 2-tensor. Now suppose $A=vv^T$ for an $\mathbb{R}^n$ vector, $v$. I want to rewrite the object ...
1
vote
2answers
66 views

Harmonic number identity

I search for an elementary proof of the following identity: $$ \sum_{i=1}^{n-k} \frac{(-1)^{i+1}}{i}\binom{n}{i+k}=\binom{n}{k}\left(H_n-H_k\right) $$ I have found the following proof: $$ ...
0
votes
1answer
22 views

Evaluating limits of exponential functions

I am just wondering how to evaluate these limits. I am aware that the method I said is not mathematically acceptable as we do not have $\infty$ as powers etc. but I just want to see whether that make ...
1
vote
3answers
26 views

A question related to harmonic numbers

For $n \geq 1$ fixed, I want to know how to compute the double sum \begin{equation} \sum_{i=1}^{n-1} \sum_{j=i+1}^n \frac{1}{i \cdot j}. \end{equation} In particular, can we say anything about the ...
1
vote
1answer
34 views

Computing the limit of a summation of sequence

How to compute the limit $$\lim _{n\rightarrow \infty }\left( \dfrac {1}{\sqrt {n^2+1^2}}+\dfrac {1}{\sqrt {n^2+2^2}}+\cdots+\dfrac {1}{\sqrt{n^2+n^2}}\right)$$ The answer is $$\ln ( \sqrt{2}+1)$$ ...
1
vote
3answers
52 views

Evaluation of $\int \frac{x^4}{(x-1)(x^2+1)}dx$

Evaluation of $\displaystyle \int \frac{x^4}{(x-1)(x^2+1)}dx$ $\bf{My\; Try::}$ Let $$\displaystyle I = \int\frac{x^4}{(x-1)(x^2+1)}dx = \int \frac{(x^4-1)+1}{(x-1)(x^2+1)}dx = \int\frac{(x-1)\cdot ...
4
votes
5answers
91 views

How to integrate $\int_{0}^{1}\ln\left(\, x\,\right)\,{\rm d}x$?

I encountered this integral in the quantum field theory calculation. Can I do this: $$ \left. \int_{0}^{1}\ln\left(\, x\,\right)\,{\rm d}x =x\ln\left(\, x\,\right)\right\vert_{0}^{1} ...
0
votes
1answer
22 views

Limits approaching from both sides go to infinity

Suppose that $\lim_{x \to a} f(x) = \infty$. Prove that we then have $\lim_{x \to a^+} f(x) = \infty$ and $\lim_{x \to a^-} f(x) = \infty$ from the definitions using epsilon-delta methods.
4
votes
7answers
166 views

Show that $\int_0^\infty \frac{\sin (\lambda x)}{e^x} \, \mathrm dx =\frac{\lambda}{1+{\lambda^2}}$

$$\int_0^\infty \frac{\sin (\lambda x)}{e^x} \, \mathrm dx =\frac{\lambda}{1+{\lambda^2}}$$ My intuition telling me there might be an $\arctan$ coming up, but I don't know how to do this ...
0
votes
0answers
14 views

How should i go about proving an expression of this kind?

Lets say i have a complete bell polynomial that is equal to a summation such that $$ B_n(d_1,d_2,\cdots,d_n) = \sum_{k=0}^{n}[g(x)^{-k} h(k)] $$ Where $d_n = \frac{d^n}{dx^n}[f(x)\ln(g(x)]$ and ...
1
vote
1answer
25 views

Critical points of an integrated function

Let $$F(x)=-\int_{0}^{x^2}\frac{2}{3+e^t}dt$$ Find all critical points of $F(x)$ and determine whether they are minima, maxima or points of inflection. Prove that $F(300)>F(310)$. First I ...
1
vote
1answer
28 views

Great books on all different types of integration techniques

It's coming up to Christmas so I can ask to have all the books I can't afford from begrudging relatives! I'm really interested (mainly from looking at some of the answers cleo and other fantastic ...
41
votes
5answers
2k views

Calculating the integral $\int_{0}^{\infty} \frac{\cos x}{1+x^2}\mathrm{d}x$ without using complex analysis

Suppose that we do not know anything about the complex analysis (numbers). In this case, how to calculate the following integral in closed form? $$\int_0^\infty\frac{\cos\;x}{1+x^2}\mathrm{d}x$$
4
votes
2answers
70 views

How to do integration of this?

$$\int_0^\infty\frac{x \sin x }{(x^2 + a^2)(x^2 + b^2)}dx\quad\quad a > b > 0$$ I have no idea how to compute this. Any help would be greatly appreciated.
0
votes
1answer
24 views

Big-Omega proof using L'Hopital's Rule?

Prove or disprove: $15n^2$ is in $\Omega(3 \times 2^n)$ So we'd have to prove or disprove this statement: $$ \exists c \in\mathbb{R}^+,\,\exists B\in\mathbb{N}, \forall n \in\mathbb{N}, n ≥ B ...
-1
votes
4answers
51 views

Calculating $\lim_{x\to\infty}\sqrt{x^2 + 5x} - x$ [on hold]

How to evaluate following limit $$\lim_{x\to\infty}\sqrt{x^2 + 5x} - x$$ I'm not sure how to factor and solve.
1
vote
2answers
25 views

Show a limit of a bounded function is 0 then solving the integral

$B=\{x^2+y^2\le1\}$, & for all $\delta>0$, there is $B_\delta=\{x^2+y^2\le\delta\}$. $f$ is a continuous function and $\|\nabla f\|\le1$ on $B$, and suppose $\frac{\partial^2f}{\partial ...
0
votes
0answers
18 views

Converting a word problem to algebra

This is a forming of an equation, which I haven't been able to get my head around. I have a worked solution to this problem. Question: For $x\in\mathbb{R}^m$ and $\epsilon>0$, show that ...
1
vote
0answers
30 views

Examples using LIATE rule for Integration by parts

Popular textbook contain many examples of integrals which can be computed by parts using the LIATE Rule. However there is almost no example of the case LI, that is a logarithmic function times an ...
3
votes
4answers
45 views

Help me, a doubt $f(x)=\cot^{-1} \frac{1-x}{1+x}$

I have a doubt $$f(x)=\cot^{-1} \frac{1-x}{1+x}$$ $$f´(x)=\frac{1}{(\frac{1-x}{1+x})^2}\cdot\frac{(-1)(1+x)-(1-x)}{1+\frac{(1-x)^2}{(1+x)^2}}$$ mm this could to be really easy but I do not ...
0
votes
1answer
17 views

Coordinates change

Lets say our vehicle is moving at $50\frac{km}{h}$ ($31 mph$) for $15$ seconds only on $X$ or $Y$-axis (also on both). How much does the Geographic coordinates change in that time? I googled and ...
8
votes
3answers
150 views
+50

Why solutions of $y''+(w^2+b(t))y=0$ behave like solutions of $y''+w^2y=0$ at infinity

Assume $w>0$ and $b(t)$ be continuous on $[0,+\infty)$ and $\int_0^\infty |b(t)| dt <\infty$ show that $y''+(w^2+b(t))y=0$ has solution $\phi(t)$ such that $$\lim_{t\to\infty} ...
3
votes
2answers
22 views

Finding a generalized form for this series

While i was just playing around with series i came across this one, $$ S = \sum_{k=1}^\infty[\frac{k}{k-\frac{1}{2}}+\frac{k-\frac{1}{2}}{k}-\frac{k+\frac{1}{2}}{k} - \frac{k}{k-\frac{1}{2}}] $$ ...
4
votes
2answers
34 views

If $\lim_{x\to\infty} [f(x+1)-f(x)] =l$ then $\lim_{ x\to\infty}f(x)/x =l$ ($f$ is continuous)

Prove that if $f$ is continuous on $\mathbb R$ and $$\lim_{x \to +\infty} [f(x+1)-f(x)] = l,$$ then $$\lim_{x\to +\infty} f(x)/x =l.$$ So I've been trying for hours to use the series ...
1
vote
2answers
60 views

I Don't Understand This Arc Length Formula

I'm taking the following from my Stewart's Calculus 7E. This is a introductory section of finding arc length. My Problem I follow what they're saying. If we approximate portions of the curve using ...
3
votes
1answer
17 views

Taylor's Theorem For Error Approximation

I'm trying to evaluate a function $f(t)$ with a given $t$ value to within 10$^{-5}$. So, if I use Taylor's Theorem : $f(a)+f′(a)1!(x−a)+f′′(a)2!(x−a)2+⋯+f(n)(a)n!(x−a)n.$ Would my $t$ value = $a, ...
0
votes
2answers
37 views

Is this a valid proof of the Quotient rule?

In an emergency in high-school, I once derived the quotient rule from the chain and product rules. I now wonder whether this was actually a valid proof. I reconstructed it as well as I could remember: ...
1
vote
2answers
42 views

Proving Polynomial is Analytic

If a function $f$ at $x = a$ equals it's Taylor Series, $f$ is said to be analytic. So, if I were given a polynomial $p(x) = \sum_{n=0}^{200}{a_nx^n}$, and trying to prove that $p(x)$ was analytic ...
0
votes
1answer
61 views

Contradiction proof for a limit law $f(x) \le g(x)$

Suppose that $f(x) \le g(x)$ for all $x$. Prove that $\displaystyle \lim_{x \to a} f(x) \le \lim_{x \to a} g(x)$, provided these limits exist. I posted a similar question, but this is a different ...
0
votes
1answer
42 views

Proof by using mean value theorem

Let $f(x)$ be a function such that it is continuous and differentiable Show that $(f(a)+f(b))/2 = f((a+b)/2) + (b-a)((f'(a)+f'(b))/2) + ((b-a)^2 / 12)f''(c)$ I have tried many different approaches ...
17
votes
2answers
182 views

Prove $\gamma_1\left(\frac34\right)-\gamma_1\left(\frac14\right)=\pi\,\left(\gamma+4\ln2+3\ln\pi-4\ln\Gamma\left(\frac14\right)\right)$

Please help me to prove this identity: $$\gamma_1\left(\frac{3}{4}\right)-\gamma_1\left(\frac{1}{4}\right)=\pi\,\left(\gamma+4\ln2+3\ln\pi-4\ln\Gamma\left(\frac{1}{4}\right)\right),$$ where ...
6
votes
6answers
277 views

Calculus Question: $\int_0^{\frac{\pi}{2}}\tan (x)\log(\sin x)dx$

Can anyone help me to find $\int_0^{\frac{\pi}{2}}\tan (x)\log(\sin x)dx$? Any help would be appreciated. Thanks in advance.
26
votes
4answers
549 views
+500

Evaluating $\int_0^1 x \tan(\pi x) \log(\sin(\pi x))dx$

What starting point would you recommend me for the one below? $$\int_0^1 x \tan(\pi x) \log(\sin(\pi x))dx $$ EDIT Thanks to Felix Marin, we know the integral evaluates to ...
2
votes
6answers
51 views

Compute $\lim_{x \to 0} \frac{\sin(x)+\cos(x)-e^x}{\log(1+x^2)}$.

Question: Compute $\lim_{x \to 0} \frac{\sin(x)+\cos(x)-e^x}{\log(1+x^2)}$. Attempt: Using L'Hopital's Rule, I have come to $$ \lim_{x \to 0} \frac{\cos(x)}{2x} - \lim_{x \to 0} ...
2
votes
1answer
32 views

Understanding a step in a double series proof

I'm really confused, how do they get from the first line to the second line ? $$\begin{align*} ...
3
votes
2answers
39 views

Prove that there exists $c \in \mathbb{R}$ such that $f'(c)=0$ if $ \lim_{x\to-\infty}f(x) =\lim_{x\to\infty}f(x) = 0 $.?

Suppose that $f$ is differentiable on $ \mathbb{R} $ and $ \lim_{x\to-\infty}f(x) =\lim_{x\to\infty}f(x) = 0 $. Prove that there exists $c \in \mathbb{R}$ such that $f'(c)=0$ . and can I extending ...
6
votes
3answers
221 views

Evaluation of $\int_0^1 \frac{x^2}{2(2-x^2)(1+x^2) + 3\sqrt{(2-x^2)(1+x^2)}}\,dx$

How does one evaluate the following integral? $$\int_0^1 \frac{x^2}{2(2-x^2)(1+x^2) + 3\sqrt{(2-x^2)(1+x^2)}}\,dx$$ This is a homework problem and I have been evaluating this integral for hours yet ...
0
votes
1answer
16 views

Velocity Vector Parallel To xy Plane

Particle moves by equations: $x\left(t\right)=5t^2,\:y\left(t\right)=10t,\:z\left(t\right)=5t^2-40t$ a) For what values of $t$ is the velocity vector parallel to the $xy$ plane Attempt: When $ t=8$ ...
-1
votes
2answers
40 views

Problem in Spivak Calculus [on hold]

Suppose that $f$ and $g$ are differentiable functions that satisfy $\int_{0}^{f(x)}f(t)g(t)dt=g(f(x))$. Show that $g(0)=0$. Thank you.
0
votes
2answers
22 views

Procedure to find a level curves

I'm having trouble finding level curves. What's the procedure? In this case, for example: $z=x^2+y^2=k$ $\hookrightarrow y=\sqrt(k-x^2)$ Then I sketch this based on knowing the formulas of ...
2
votes
3answers
34 views

Finding $\lim_{x\rightarrow 0}\frac{x}{2}\sqrt{\frac{1+\cos(x)}{1-\cos(x)}}$

We know that $$\sqrt{\frac{1-\cos(x)}{1+\cos(x)}}=\tan({x}/{2})$$ so we can change the above function to another form as follow ...
1
vote
1answer
23 views

function of three variable is even than $f(a,b,c)=f(|a|,|b|,|c|)$

I used in the proof of Hlawka's Inequality you can find the link here Hlawka's Inequality that's if i have function of three variable is even in each variable, so that : $$f(a,b,c)=f(|a|,|b|,|c|)$$ ...
2
votes
1answer
252 views

Differentiating a convolution integral

I'm trying to turn the integro-differential equation $\phi'(t) + \phi(t) = \int_0^t \sin{(t - \xi)} \, \phi(\xi) \, \mathrm{d} {\xi}$ into the differential equation $\phi'''(t) + \phi''(t) + ...
1
vote
1answer
29 views

Show that for $x\geq 1$ and $c>0$, show that $\dfrac{1}{x}\leq x^{c-1}$

I've done part a, and I presume I'm somehow meant to use it to do part b, but I have no idea how to do it. for part (a): $a=1, b=n, f(x)=\dfrac{1}{x}$
2
votes
1answer
52 views

Sequences for that $\sum_{n} \frac{1}{x_n}$ is divergent and $\sum_{n} \frac{1}{x_n \ln x_n}$ is convergent

We will denote with $(x_n)$ a given sequence and we introduce the following two series. $$S^* = \sum_{n} \frac{1}{x_n} \quad \text{and} \quad S_* = \sum_{n} \frac{1}{x_n \ln x_n}.$$ We know that if ...
17
votes
3answers
533 views

Closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$.

What is the closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$? We can use Fourier series of $e^{-bx}$ ($|x|<\pi$) to evaluate $\sum_{n=-\infty}^{\infty}\frac{1}{n^2+b^2}$. But this ...
1
vote
1answer
14 views

calculate $\lim_{n\to\infty}\int_{[0,\infty)} \exp(-x)\sin(nx)\,\mathrm{d}\mathcal{L}^1(x)$

We've had the following Lebesgue-integral given: $$\int_{[0,\infty)} \exp(-x)\sin(nx)\,\mathrm{d}\mathcal{L}^1(x)$$ How can you show the convergence for $n\rightarrow\infty$? We've tried to use ...
1
vote
1answer
7 views

Show that $4 - Un+1 < 1/2(4 - Un)$

Let Un be a sequence such that : U0 = $0$ ; Un+1 = $sqrt(3Un + 4)$ We know (from a previous question) that Un is an increasing sequence and Un < $4$ Show that $4$ - Un+1 <(or =) 1/2(4-Un) I ...