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

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3
votes
0answers
13 views

How to derivative the linear equation of matrix

I have the equation as $$F(w,x)=\sum_{i=1}^{N}\int_{x \in \Omega} \left ( Y(x)-w^TA(x)\right)^2u_i(x)dx$$ In which, $w$ is column vector that independent on $x$, denotes $w=[w_1,w_2...,w_M]^T$ $A$ ...
0
votes
0answers
15 views

check whether the function is increasing or decreasing or neither increasing nor decreasing

If $x\in[0,\pi]$, then function $$f(x)=x\sin x+\cos^2 2x(\cos (x^2))$$ is increasing or decreasing or neither increasing nor decreasing.
2
votes
1answer
64 views

How to find the pointwise limit of $(f_n)$

For $x \in [0, \pi/2]$, if $$f_n(x) = \frac {nx} {1+n\sin(x)}$$ how do you find the pointwise limit of $(f_n)$ ?
5
votes
5answers
1k views

Math Subject GRE 1268 Question 55

If $a$ and $b$ are positive numbers, what is the value of $\displaystyle \int_0^\infty \frac{e^{ax}-e^{bx}}{(1+e^{ax})(1+e^{bx})}dx$. A: $0$ B: $1$ C: $a-b$ D: $(a-b)\log 2$ E: ...
1
vote
3answers
75 views

Proving $(1 + \frac{1}{n})^n < n$ for natural numbers with $n \geq 3$.

Prove with induction on $n$ that \begin{align*} \Bigl(1+ \frac{1}{n}\Bigr)^n < n \end{align*} for natural numbers $n \geq 3$. Attempt at proof: Basic step. This can be verified easily. Induction ...
5
votes
3answers
67 views

Closed-form of $\int_0^1 x^n \operatorname{li}(x^m)\,dx$

I've conjectured, that for $n\geq0$ and $m\geq1$ integers $$ \int_0^1 x^n \operatorname{li}(x^m)\,dx \stackrel{?}{=} -\frac{1}{n+1}\ln\left(\frac{m+n+1}{m}\right), $$ where $\operatorname{li}$ is the ...
124
votes
2answers
8k views

Evaluate $ \int_{0}^{\frac{\pi}{2}}\frac1{(1+x^2)(1+\tan x)}\,\mathrm dx$

Evaluate the following integral$$\int_{0}^{\Large\frac{\pi}2}\frac1{(1+x^2)(1+\tan x)}\,\mathrm dx.$$ My Attempt: Let $$\tag1 I = \int_{0}^{\Large\frac{\pi}{2}}\frac{1}{(1+x^2)(1+\tan ...
1
vote
1answer
36 views

Matrix integration by parts

It seems to me that the integration by parts rule carries over simply to the matrix case. This can be seen by applying: $(AB)' = A'B + AB'$ and then integrating for square (time dependent) complex ...
0
votes
0answers
15 views

How to make sense of this contour graph?

Recently, I have been studying the paper "Estimating the basic reproductive ratio for the Ebola outbreak in Liberia and Sierra Leone", published in Infectious Diseases of Poverty. I came across the ...
-1
votes
1answer
32 views

$[\tan x]^2+\tan x-a$

What is the number of integral values of $a$,$a\in(6,100)$ for which the equation $[\tan x]^2+\tan x-a=0 $ has real roots, where [.] denotes greatest integer function. My try:$[\tan x]^2+\tan ...
0
votes
0answers
14 views

An inequality involving supremum and integral 2

In the following post An inequality involving supremum and integral, there was discussed the inequality $$ \sup_{r<t<\infty}g(t)\leq C\int_{r}^{\infty}g(t)\frac{dt}{t}, $$ where positive ...
2
votes
1answer
52 views

“Mean value like” problem.

Let $f:\mathbb{R} \longrightarrow \mathbb{R}$ be differentiable, take $a<a'<b<b'$. Prove that there exists $c<c'$ such that $$\frac{f(b)-f(a)}{b-a}=f'(c) \quad and \quad ...
1
vote
4answers
42 views

How does this algebraic trick regarding partial fraction works?

Suppose I have to evaluate the integral $$\int \frac{x}{(x-1)(2x+1)(x+3)} \, dx $$ I write it as $$\frac{a_1}{x-1} +\frac{a_2}{2x+1} +\frac{a_3}{x+3}$$ where $a_1$, $a_2$, $a_3$ are constants. once I ...
7
votes
1answer
76 views

What exactly IS a line integral?

As what happens in many math courses, a topic is learned without truly learning what one is doing. For me, this is line integrals. I can do them well, I just never truly learned what exactly I was ...
0
votes
2answers
48 views

Differential equation $f'''(x)=-f(x)$ with restriction using power series

Using power series, Prove the existence of a $C^3$ function (continuously differentiable 3 times) $f:\mathbb{R} \to \mathbb{R}$ such that $f'''(x)=-f(x)$ $\forall x \in \mathbb{R}$ and ...
0
votes
2answers
39 views

Rotate a point on a circle with known radius and position

Having a circle $\circ A(x_a, y_a)$ of radius $R$ and a point on the circle $B(x_b, y_b)$, how can we rotate the point with a known angle $\alpha$ (radians or degrees, it doesn't really matter) on the ...
2
votes
2answers
106 views

Integration of $\int \frac{(1 + x)\sin x}{(x^2 +2 x)\cos^2 x-(1 + x)\sin2x}dx$

The integral is $$\int \dfrac{(1 + x)\sin x}{(x^2 + 2x)\cos^2 x-(1 + x)\sin2x}dx.$$I've tried the problem by first multiplying both the numerator and denominator by $\sec^2 x$ but couldn't do justice. ...
0
votes
3answers
87 views

Spivak's Calculus, chapter 1 problem 19 (inequalities)

I'm having trouble with problem 1-19 in Spivak's Calculus. I have to prove that if $|x-x_0| < \frac{\epsilon}{2} $ and $ |y-y_0| < \frac{\epsilon}{2} $ then $ |(x-y)-(x_0-y_0)| < \epsilon $. ...
1
vote
3answers
163 views

Two similar method to calculate one equation get different answer

Method1:$$\lim_{x\rightarrow0}({\frac{e^x+xe^x}{e^x-1}}-\frac1x)=\lim_{x\rightarrow0}({\frac{e^x+xe^x}{x}}-\frac1x)=\lim_{x\rightarrow0}(\frac{e^x+xe^x-1}{x})=\lim_{x\rightarrow0}(2e^x+xe^x)=2$$ ...
0
votes
1answer
44 views

Find the integral of $x(x^2+4)^5$? [on hold]

Please do show me step by step way of integrating $x(x^2+4)^5$. Please do provide me with the formulas, if any. In my text-book, the answer is given as: $(\frac{1}{12})(x^2+4)^6 + k$ But I don't know ...
3
votes
5answers
59 views

What rule can I use to compute $\frac{d^{107}}{dx^{107}} \sin x$?

Did I miss something in my calculus class? I don't remember anything concerning this type of problem: Compute $$\frac{d^{107}}{dx^{107}} \sin x.$$ So what is the rule here?
2
votes
2answers
71 views

Can this integral diverge?

Let $f:\mathbb{R}_+\to\mathbb{R}_+$ be a continuous non-increasing bounded function, and suppose $$\limsup_{x\to +\infty}\frac{f(x)}{f(2x)} = +\infty. $$ Can $\int_0^{+\infty} f(x)dx$ diverges? ...
3
votes
2answers
108 views

Rearranging Pokemon Experience Formula to make Level the Subject

As the title suggests, I am trying to rearrange some of the formulas for calculating experience based on level to be the other way around (calculating level based on experience). I am having trouble ...
1
vote
1answer
69 views

Are there expansions of the expression $(a+b)^{1/n}$? [duplicate]

Is there an expansion of the expression in the bracket such as $$ \sqrt{a + b} = (a + b)^{1/2}$$ If not do you know of a method that lets us solve such expression and ones with higher roots?
0
votes
5answers
99 views

How do I solve $5^x = x^2$? [on hold]

How do I solve the equation in the title? This was in a Calculus text book that did not show the answer to this.
0
votes
2answers
100 views

Math Subject GRE 1268 Problem 64 Flux of Vector Field

What is the value of the flux of the vector field F, defined on $R^3$ by $F(x,y,z) = (x,y,z)$ through the surface $z=\sqrt{1-x^2-y^2}$ oriented with upward-pointing normal vector field? ...
0
votes
1answer
49 views

Number of real roots of polynomial derivative

Let $W(x)$ be a polynomial with n real roots and $P(x) = \alpha W(x) + W'(x)$. Prove that for any $\alpha \in \mathbb{R}$: $P(x)$ have at least $n-1$ real roots. I know that the degree of the ...
2
votes
3answers
383 views

Definite integrals using u substitution (verification needed)

Would someone mind verifying this? $ \int_{0}^{\ln(\pi + 1)}e^x \sin(e^x - 1) \space dx $ $ u = e^x - 1 \Rightarrow \frac{du}{dx} = e^x \Rightarrow du = e^x \space dx \Rightarrow dx = \frac{1}{e^x} ...
1
vote
1answer
33 views

How do I determine the new boundaries of $D ^* = T(D)$ when using change of variable?

I'm not quite sure how to complete this question $D$ is the region bounded by $x = 0, y=0,x+y=1$, and $x+y=4$. Using the change of variables $x = u -uv, y = uv$ and the Jacoian, evaluate the double ...
1
vote
2answers
24 views

Lagrange Multipliers: Rectangular Box

Problem Glass costs twice as much as plywood, per square meter. Use Lagrange multipliers to answer: What is the shape of the cheapest rectangular box, with 5 rectangular plywood sides and 1 ...
1
vote
2answers
48 views

For what $(a,b) \in R^+$ does $\int^\infty_b (\sqrt{\sqrt{x+a}-\sqrt{x} \vphantom{\sqrt{x}-\sqrt{x-b}}}-\sqrt{\sqrt{x}-\sqrt{x-b}})dx$ converge?

For what pairs $(a,b) \in R^+$ does this integral converge? $$ \int\limits^{\infty}_{b} \left (\sqrt{\sqrt{x+a}-\sqrt{x} \vphantom{\sqrt{x}-\sqrt{x-b}}}-\sqrt{\sqrt{x}-\sqrt{x-b}} \right)dx $$
1
vote
0answers
18 views

Finding an anti-derivable non-linear function of a Fourier partial sum

I'm working on a project where I need to compute definite integrals of the composition $\sigma(g(x))$, where $\sigma(x)$ is any non-linear amplifier/activation function and $g(x)$ is the sum of many ...
13
votes
1answer
167 views
+50

Linear differential equations of the $n$th order

$$ L(x)=x^{(n)}+a_1(t)x^{(n-1)}+\cdots +a_{n-1}(t)x'+a_n(t)x;\qquad a_1(t),a_2(t),\ldots\in C$$ $$U_j(\varphi)= \sum_{k=0}^{n-1}(M_{jk} \varphi^{k}(\alpha)-N_{jk} \varphi^{k}(\beta))= \gamma_j\quad ...
2
votes
4answers
46 views

How can I find an ODE equation from $dy/dx$

What is the ODE satisfied by $y=y(x)$ given that $$\frac{dy}{dx} = \frac{-x-2y}{y-2x}$$ I understand that I need to get it in some form of $\int \cdots \;dy = \int \cdots \; dx$, but am not sure ...
0
votes
2answers
30 views

What is the Right-Hand Limit as x approaches 0 of the nth root?

A textbook says that "One-sided limits are useful in taking limits of functions involving radicals. For instance, if is an even integer, then I need help proving this, Ive gotten as far as this ...
1
vote
1answer
46 views

Help needed with the integral of an infinite series

Can you please help me with the integral of this series? I came across it in a signal processing paper and haven't been able to figure out the solution myself. $$ ...
4
votes
0answers
78 views

Integral $\int z^2\Re(J_1(z))dz$=$\int y^{3/2} \Re \left[\frac{1}{\sqrt y} (1-e^{-y})\right]dy$

Hi I am trying to simplify and calculate the integral below. $$ I=\int x^2 \, \Re\left[{J_1(a x)}\right]dx=\frac{1}{a^3}\int z^2 \Re\left[\frac{z}{2}\sum_{k\geq 0} \frac{(-1)^k}{k!\Gamma(k+2)} ...
0
votes
0answers
24 views

Calculus: Diffeomorphisms

Problem Consider the map $(u,v)=F(x,y)=(x+x^2,y-xy)$. Take a positive number $r>0$ and let $D_r$ be the disk of radius $r$ around the origin in the $x,y$-plane, and let $U_r=F(D_r)$ be its image in ...
8
votes
4answers
1k views

10th derivative of a function

I want to find $f^{(10)}(0)$ where $f(x)=\ln(2+x^2)$. I know that it can be done "by hand", but I believe there is a smarter way. I think I should use Taylor series and the fact that ...
0
votes
3answers
604 views

The normal line intersects a curve at two points. What is the other point?

The line that is normal to the curve $\displaystyle x^2 + xy - 2y^2 = 0 $ at $\displaystyle (4,4)$ intersects the curve at what other point? I can not find an example of how to do this equation. Can ...
2
votes
1answer
49 views

Equilibrium Points Second Order Differential

Attempt: I get the system of the two first order equations (first order in $w$) by considering the different signs the first derivative takes. Problem is by equilibrium points: do I just set the ...
2
votes
3answers
42 views

Classifying a differential equation

How do I classify the following differential equation? In particular, is this differential equation "homogeneous?" $$(x^3+3y^2)dx-2xydy=0$$ Solving it is not the problem, but I don't know how ...
5
votes
3answers
121 views

Problem 7 IMC 2015 - Integral and Limit

I'm trying to solve problem 7 from the IMC 2015, Blagoevgrad, Bulgaria (Day 2, July 30). Here is the problem Compute $$\large\lim_{A\to\infty}\frac{1}{A}\int_1^A A^\frac{1}{x}\,\mathrm dx$$ ...
-4
votes
1answer
50 views

Difficult product problem $\prod \limits^{2014}_{k=1}\left( 1-\frac{1}{k^{2}} \right)$ [duplicate]

Evaluate the product $$\prod \limits^{2014}_{k=1}\left( 1-\frac{1}{k^{2}} \right)$$ Any help will appreciated!
-1
votes
1answer
58 views

Why is $f(x) = \sin(x)$ an element of $L^2(-\pi, \pi)$ not $L^2(a,b)$ [on hold]

I am having some trouble understanding why some functions are members of $L^2(\mathbb{R})$ whereas other functions are members of some restricted subset of $\mathbb{R}$ such as $(-\pi, \pi)$ Can ...
2
votes
5answers
36 views

Finding a general solution to a differential equation, using the integration factor method

Use the method of integrating factor to solve the linear ODE $$ y' + 2xy = e^{−x^2}.$$ And verify your answer I can solve the ODE as a linear equation (mulitply both sides, subsititute, reverse ...
0
votes
0answers
16 views

Differentiating composition of functions proof help

Theorem: Let $X, Y, Z$ be normed spaces and $U\subset X$, $V \subset Y$ open sets. If the function $f:U \to V$ is differentiable in $x \in U$ and function $g: V \to Z$ differentiable in $f(x)\in ...
1
vote
1answer
27 views

Mulitvariable Calculus: Vectors of tangent plane

Problem Let $M$ be the surface consisting of the points $(x,y,z)\in \mathbb{R}^3$ for which $x^2+y^3+z^4=1$. Which vectors $v\in \mathbb{R}^3$ belong to the tangent plane to M at the point ...
0
votes
2answers
25 views

Problem with understanding the application of the Intermediate Value Theorem in the proof of the Mean Value Theorem for Integrals

I am struggling to understand the last parts of this proof because I know that the IVT states that on the interval $[a,b]$ of $f$, where it is continuous, there exists a value $L$ between $f(a)$ and ...
31
votes
2answers
530 views
+500

Closed form for $\int_0^1\frac{x^{5/6}}{(1-x)^{1/6}\,(1+2\,x)^{4/3}}\log\left(\frac{1+2x}{x\,(1-x)}\right)\,dx$

I need to evaluate this integral: $$Q=\int_0^1\frac{x^{5/6}}{(1-x)^{1/6}\,(1+2\,x)^{4/3}}\log\left(\frac{1+2x}{x\,(1-x)}\right)\,dx.$$ I tried it in Mathematica, but it was not able to find a closed ...