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

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12
votes
4answers
1k views

Which methods to use to integrate $\int{\frac{x^4 + 1}{x^2 +1}}\, dx$

I have this integral to evaluate: $$\int{\frac{x^4 + 1}{x^2 +1}}\, dx$$ I have tried substitution, trig identity and integration by parts but I'm just going round in circles. Can anyone explain ...
1
vote
1answer
144 views

$ \sum_{y=1}^\infty {}_1F_1(1-y;2;-\pi\lambda c) \frac{\lambda^y}{y!} $

I am not able to solve the following sum. Can you please provide any hints ? $$ \sum_{y=1}^\infty {}_1F_1(1-y;2;-\pi\lambda c) \frac{\lambda^y}{y!} $$ Note that the 3rd parameter of the Confluent ...
0
votes
1answer
79 views

Compute limit of the sequence given by $x_1=1$, $x_{n+1}^2=\frac{x_n+3}{2}$

Let $\left(x_n\right)$ be a real sequence such that $x_1=1,x_{n+1}^2=\dfrac{x_n+3}{2},\forall n\geq 1$.Compute $\lim_{n\to+\infty} 3^n\sqrt{\dfrac{9}{4}-x_n^2}$
2
votes
3answers
395 views

Find the antiderivative of $\sqrt{3x-1} dx$

Find the antiderivative of $\sqrt{3x-1} dx$. I got $\frac{2}{3}(3x-1)^{3/2}+c$ but my book is saying $\frac{2}{9}(3x-1)^{3/2}+c$ Can some one please tell me where the $2/9$ comes from?
4
votes
4answers
782 views

How to find inverse of the function $f(x)=\sin(x)\ln(x)$

My friend asked me to solve it, but I can't. If $f(x)=\sin(x)\ln(x)$, what is $f^{-1}(x)$? I have no idea how to find the solution. I try to find ...
1
vote
1answer
35 views

Question about integrability

Let f be a continious function on [a,b] and exist a partition P of [a,b] such that $\bar{S}(f,p)=\int_a^b f(x)dx$. Prove that f is a constant function. I thought stratting assuming the claim is not ...
1
vote
2answers
89 views

Evaluate $\int \dfrac{1}{\sqrt{1-x}}\,dx$

Find $$\int \dfrac{1}{\sqrt{1-x}}\,dx$$ I did this and got $\dfrac23(1-x)^{\frac32} + c$ But a online calculator is telling me it should be $-2(1-x)^{\frac12}$ What one is on the money and if not ...
8
votes
2answers
377 views

How prove this $\sum_{n=1}^{\infty}\frac{\zeta_{2}}{n^4}=\zeta^2(3)-\frac{1}{3}\zeta(6)$

show that $$\sum_{n=1}^{\infty}\dfrac{\zeta_{2}}{n^4}=\zeta^2(3)-\dfrac{1}{3}\zeta(6)$$ where $$\zeta_{m}=\sum_{k=1}^{n}\dfrac{1}{k^m},\zeta(m)=\sum_{k=1}^{\infty}\dfrac{1}{k^m}$$ is true? because ...
0
votes
1answer
39 views

Show relation for integrals

Let $f \in C^{1}([a,b];\mathbb{R})$ and $|f'(x)-f'(y)| \le L |x-y|$ then we have $|\int_a^b f(x) dx -f(\frac{a+b}{2})(b-a)| \le L\frac{(b-a)^3}{4}$. I have troubles to show this inequality. the ...
2
votes
2answers
72 views

Show that $\sum_{n=1}^{\infty} {(-1)^n \sin(\frac{x}{n})}$ converges for $a > 0$ in $[-a,a]$

Given: $$\sum_{n=1}^{\infty} {(-1)^n \sin(\frac{x}{n})}$$ We need to show that it converges in $[-a,a]$ $\forall a >0$. Now what I figured out: This is a function series summation. we need ...
1
vote
1answer
39 views

Calculating the Riemann sum $\lim_{n \to \infty} { \sum_{k=1}^{n} { (\frac{nk-1}{n^3})\sin(\frac{k}{n}) } }$ [duplicate]

We need to calculate this: $$\lim_{n \to \infty} { \sum_{k=1}^{n} { (\frac{nk-1}{n^3})\sin(\frac{k}{n}) } }$$ So I know this is Riemann sum. This is what I started doing: $$\sum_{k=1}^{n} ...
0
votes
1answer
107 views

Changing order of derivatives

I would like to rewrite the following expression $$\frac{d^i}{dx^i}\left\{f(x)\left[\frac{d^jf(x)}{dx^j}\right]\left[\frac{d^kf(x)}{dx^k}\right]\right\}$$ into the form $$D f(x)^3,$$ with $D$ ...
4
votes
4answers
1k views

Proving one function is greater than another

How can I prove $f(x)$ $>$ $g(x)$ for all $x > 0$ given $f(x) = (x+1)^{2}$ and $g(x) = 4qx$ where $q$ is a constant in $(0, 1)$? My approach was to show that $(x+1)^2 > 4qx$ for the interval ...
2
votes
1answer
91 views

Does $\lim\limits_{n\to\infty}\frac{\sum_{i=1}^{n^2} 1}{\sum_{i=1}^n i}$ result into 2?

How does the following limit $$\lim\limits_{n\to\infty}\frac{\displaystyle\sum_{i=1}^{n^2} 1}{\displaystyle\sum_{i=1}^n i}$$ result into 2? Something like: ...
1
vote
3answers
107 views

How does $\lim\limits_{n\to\infty} \sum_{i=1}^{n^2} 1$ result into 1+3+5… [duplicate]

How can $\lim\limits_{n\to\infty} \sum_{i=1}^{n^2} 1$ result into 1+3+5...? Why odd numbers?
0
votes
1answer
54 views

The sum of the integration of g and $g^{-1}$

Let $g$ be a strictly increasing continuous function mapping $[a,b]$ onto $[A,B]$, and, as usual, let $g^{-1}: [A,B] \to [a,b]$ denote its inverse function. Use geometric insight to visualize the ...
2
votes
3answers
82 views

Explain the 1 + 2 + 3 in $ \frac{1 + 1 + 1 + \cdots}{1 + 2 + 3 + \cdots} = \lim_{n \to \infty} \frac{1}{(n+1)/2} $

$$ \frac{1 + 1 + 1 + \cdots}{1 + 2 + 3 + \cdots} = \lim_{n \to \infty} \frac{1}{(n+1)/2} = 0 $$ If $n$ goes to infinity, we can image that a bit by taking a very big number. Like $1.000.000.000$ ...
0
votes
1answer
70 views

How can $\frac{1+1+1+\ldots}{2+2+2+\ldots} be \frac{(1+1)+(1+1)+\ldots}{2+2+2+\ldots} = 1$?

$\frac{1}{2} = \frac{1+1+1+\ldots}{2+2+2+\ldots} = \frac{(1+1)+(1+1)+\ldots}{2+2+2+\ldots} = 1$ says user Karolis Juodelė. Why does it get 7 votes up for this comment? I guess that 1/2 is not 1. See: ...
2
votes
3answers
65 views

derive using the chain rule

Given the polinomyal $f(x)=\frac{x^3}{(4-x^2)^3}$ find $f'(x)$ So, If I try to derive this, first I must to apply the chain rule in the denominator and then derive of the division (...) ...
6
votes
1answer
127 views

What is $\int x^re^xdx$?

Is there any simple way to get integral of $e^{x}x^{r}, r \in \mathbb{R}$? Basically I want to solve this: $$\displaystyle \int \frac{e^t(4t^2+1)}{2t \sqrt{t}}dt$$ so I will appreciate any help ...
28
votes
1answer
683 views

Closed form for $\int_0^\infty\frac{\log\left(1+\frac{\pi^2}{4\,x}\right)}{e^{\sqrt{x}}-1}\mathrm dx$

I encountered this integral in my calculations: $$\int_0^\infty\frac{\log\left(1+\frac{\pi^2}{4\,x}\right)}{e^{\sqrt{x}}-1}\mathrm ...
-1
votes
1answer
74 views

Need Help Solving Logarithmic Equation $2(5)^x=3^{x+1}$

Need Help Solve Logarithmic Equation. Thanks Gary $$2(5)^x = 3^{x+1}$$
3
votes
1answer
116 views

looking for reference or nice proof of trig lemma

Math people: I am looking for a reference or a nice proof of the following fact. I have proven it myself, but my proof is messy: let $\theta \in (0,1]$ and $\alpha \in (0, \frac{1}{2}\theta^2]$. ...
4
votes
1answer
219 views

Estimating the sum $\sum_{k=2}^{\infty} \frac{1}{k \ln^2(k)}$

By integral test, it is easy to see that $$\sum_{k=2}^{\infty} \frac{1}{k \ln^2(k)}$$ converges. [Here $\ln(x)$ denotes the natural logarithm, and $\ln^2(x)$ stands for $(\ln(x))^2$] I am ...
5
votes
3answers
470 views

integration by substitution, using $\;t = \tan \left(\frac 12 x\right)$

$\displaystyle\int_0^\frac{\pi}{2}\frac{1}{2-\cos x} \, dx$ using the substitution $t=\tan\frac{1}{2}x$ $x=2\tan^{-1}t$ $\dfrac{dx}{dt}=\dfrac{2}{1+t^2}$ $dx=\dfrac{2}{1+t^2}\,dt$ ...
1
vote
0answers
53 views

Product of Fourier integrals

I am interested in solving the following integral: \begin{equation} I =\int dx_{3}\psi^{\star}(x_{3})\int dx_{1}\psi(x_{1})\int dq_{1}X(q_{1})e^{iq_{1}(x_{3}-x_{1})}\int dx_{2}\psi(x_{2})\int ...
2
votes
3answers
66 views

Help evaluating $\int_0^\infty \frac{1}{x^{1/2}(x+1)}dx$

I began solving this with U sub and partial fractions...first for $x^{1/2}$ and then for $x+1$ but neither of those methods got me the answer of $\pi$. I know the indefinite integral should be ...
3
votes
3answers
28 views

Showing that a function has a certain absolute minimum.

Suppose we have the function $$f(x) = \frac{x}{p} + \frac{b}{q} - x^{\frac{1}{p}}b^{\frac{1}{q}}$$ where $x,b \geq 0 \land p,q > 1 \land \frac{1}{p}+\frac{1}{q} = 1$ I am trying to show that $b$ ...
1
vote
2answers
69 views

convergence of series with absolute value

prove or show false: if $\sum_{n=1}^{\infty}\left |a_{n} \right |$ converges, then $\sum_{n=1}^{\infty}\frac{n+1}{n}a_{n}$ converges as well. Thank you very much in advance, Yaron.
0
votes
2answers
98 views

Post-Uni Calculus/Probabilities Book Suggestion

I have a Computer Science Background, recently graduated and I would like to refresh/improve my knowledge about probabilities and statistics (also calculus). The priority is probabilities and ...
0
votes
1answer
143 views

what does “in wide sense” mean?

I came across the statement "the sequence increases(in wide sense)". So my doubt is what does author mean by wide sense?I came across this in number theory book
1
vote
2answers
392 views

A less known definition of the definite integral of a continuous function

The definite integral of a continuous function can be defined using the bounded monotone sequence property: see Osgood's Functions of Real Variables, p.110. (link to full book) (screenshots: page ...
3
votes
1answer
75 views

For which values of $\alpha \in \mathbb R$ two improper integrals converge

Question is: For which values of $\alpha \in \mathbb R$ the following improper integrals converge: a.$$\int_0^1\!\left|\ln(x)\right|^\alpha\,dx$$ ...
4
votes
1answer
162 views

question about Riemann zeta $\zeta (0)$ [duplicate]

i know that $$\zeta (m)=\sum_{n=1}^\infty n^{-m}$$ so $$\zeta (0)=\sum_{n=1}^\infty n^0=1+1+1+1+1+1+\cdots=\infty $$ but actually $$\zeta (0)=-0.5$$ where is the wrong please help thanks ...
2
votes
1answer
46 views

Proving that length of a curve is $\infty$

Let $f$ be a differentiable and continious function in $(0,1]$ and $lim_{x\to 0^+}f(x)=\infty$. Prove that the length of the curve on (0,1] is $\infty$. Steps I tried: $L=\int_0^1 ...
2
votes
0answers
212 views

What's the difference between $\frac{\delta}{dt}$ and $\frac{d}{dt}$?

I have read the few questions on calculi notation, particularly the notations on partial and total derivatives. My question seems to have not been answered, or at least not brought to my attention. If ...
0
votes
1answer
76 views

Application of derivative - how to calculate change in error

Problem : If the error committed in measuring the radius of the circle is $0.05\%$ then find the corresponding error in calculating the area. Solution : Let the error can be denoted by $\delta r = ...
5
votes
1answer
120 views

Having trouble using eigenvectors to solve differential equations

The question asked to solve $$\frac{dx}{dy} = \begin{pmatrix} 5 & 4 \\ -1 & 1\\ \end{pmatrix}x$$ ,where $$ x = \begin{pmatrix} x_1 \\ x_2 \\ \end{pmatrix}$$ I went ...
2
votes
1answer
83 views

Prove that $(1+1/x)^x$ is concave for $x>0$

From the graph it looks like $(1+1/x)^x$ is concave for $x>0$. But in this post, I can only prove that it is concave for $x\ge 1$. It is of interest to see a proof for $x>0$.
2
votes
0answers
57 views

Basic Calculus Question Using Mean Value Theorem

How does one use the Mean Value Theorem to show: $\frac{1 - e^{-\lambda}}{\lambda} \leq min(1, \frac{1}{\lambda})$
17
votes
3answers
312 views

Closed form for n-th anti-derivative of $\log x$

Is it possible to write a closed-form expression with free variables $x, n$ representing the n-th anti-derivative of $\log x$?
0
votes
2answers
248 views

$\sum \frac{\ln(n)}{\sqrt{n^5}}$ test for convergence

Let $\sum a_{n}=\sum \frac{\ln(n)}{\sqrt{n^5}}$. To find if the serie is convergence or not, I had some difficult on finding the proper serie to test the given one. After some work around, I found ...
0
votes
2answers
61 views

Integral of a vector field

I'm trying to evaluate the following integral: $ \int_C(y+\sin x) dx +(z^2+\cos y)dy+(x^3)dz$ Where $C$ is the curve: $c(t) = (\sin t, \cos t, \sin 2t) $. Note that $C$ lies on the surface ...
1
vote
3answers
436 views

The population of a certain bacteria can multiply threefold in 24 hours. If there are 500 bacteria now, how many will there be in 96 hours?

The population of a certain bacteria can multiply threefold in 24 hours. If there are 500 bacteria now, how many will there be in 96 hours? I figured out this bacteria $=500(3)^{96/24}$ but then my ...
1
vote
2answers
83 views

question on summation?

Please, I need to know the proof that $$\left(\sum_{k=0}^{\infty }\frac{n^{k+1}}{k+1}\frac{x^k}{k!}\right)\left(\sum_{\ell=0}^{\infty }B_\ell\frac{x^\ell}{\ell!}\right)=\sum_{k=0}^{\infty ...
3
votes
2answers
89 views

Euler lagrange equation is a constant

I'm working through exercises which require me to find the Euler-Lagrange equation for different functionals. I've just come across a case where the Euler Lagrange equation simplifies to $$1=0.$$ ...
0
votes
1answer
165 views

Right Triangles and Lagrange Multipliers

Suppose that you have a right triangle $a^2+b^2=c^2$ with integral sides. Given a perimeter $p=a+b+c$, how can you use Lagrange multipliers to determine the maximum length of $a$?
5
votes
5answers
731 views

Limit as $x$ approaches $1$ from the right of $\frac{1}{\ln x}-\frac{1}{x-1}$

$$ \lim_{x\rightarrow 1^+}\;\frac{1}{\ln x}-\frac{1}{x-1} $$ So I would just like to know how to begin to solve this limit, or what topic does this problem fall under so that I can search for ...
6
votes
1answer
73 views

What is the limit of the multidimensional integral?

What is the limit of the integral $$\int_{[0,1]^n}\frac{x_1^5+x_2^5 + \cdots +x_n^5}{x_1^4+x_2^4 + \cdots +x_n^4} \, dx_1 \, dx_2 \cdots dx_n$$ as $n \to \infty ?$
1
vote
1answer
219 views

Differentiation problem of power to infinity by using log property

Problem: Find $\frac{dy}{dx}$ if $y =\left(\sqrt{x}\right)^{x^{x^{x^{\dots}}}}$ Let ${x^{x^{x^{\dots}}}} =t. (i)$ Taking $\log$ on both sides $ \implies {x^{x^{x^{\dots}}}}\log x = \log t$ This ...