1
vote
0answers
39 views

From 2 to 3 dimensions: integrating a force along a contour/surface.

I am studying the following problem: Consider a closed contour $\mathcal{C}$ in $\mathbb{R}^2$ defined by $r(\theta)$ where $\theta\in[0,2\pi)$ and $r(0)=r(2\pi)$ (let the center to be zero for ...
0
votes
3answers
106 views

Proving Riemann Sums via Analysis

Exercise $\bf 5.1.7$: Suppose $f:[a,b]\to\Bbb R$ is Riemann integrable. Let $\epsilon\gt0$ be given. Then show that there exists a partition $P=\{x_0,x_1,\ldots,x_n\}$ such that if we pick any set ...
2
votes
3answers
166 views

Proof that $\int \frac{1}{x}$ is $\ln(x)$

When I was learning Calculus AB and Calculus II/III at my high school, I noticed that our textbooks never gave a full fundamental proof that $\int \frac{1}{x}$ is $\ln(x)$, and rather said that when ...
0
votes
0answers
46 views

algebraic proof of integration

You can prove integration in this way with algebra and number theory for the problem y=2x^2+1 or y=ax^2+c: $$S_n=\frac{2b^3k^2}{n^3}+\frac{b}{n}....... ...
0
votes
0answers
28 views

U-substitution proof by partitions

I am trying to prove integration by substitution, i.e. for $f:[c,d] \rightarrow \mathbb{R}$ continuous and $\phi: [c,d] \rightarrow [a,b]$ continuous on $[c,d]$ and $C^1$ on $(c,d)$. Then define ...
1
vote
2answers
41 views

Show a double-sided infinite integral of $\sin(x+b)$ exists iff $b=n\pi$

More formally: Show that $$\lim_{a\rightarrow \infty} \int_{-a}^a \sin(x+b)$$ exists if and only if $b=n\pi$ for some $n \in \mathbb{Z}$. I get the intuition fine. The function is just a horizontal ...
4
votes
0answers
79 views

Proving u-substitution the hard way — use only definition of integration with partitions

I am trying to prove integration by substitution, i.e. for $f:[c,d] \rightarrow \mathbb{R}$ continuous and $\phi: [c,d] \rightarrow [a,b]$ continuous on $[c,d]$ and $C^1$ on $(c,d)$. Then define ...
0
votes
0answers
32 views

Show if this is integrable (defined 1 on rationals, 0 else)

Define $f: [0,1] \rightarrow \mathbb{R}$ as $f(x) = \begin{cases} 0 & x \in \mathbb{Q} \\ 1 & x \notin \mathbb{Q} \end{cases}$ Find $\underline{\int_0^1f}$ and $\overline{\int_0^1f}$. Is ...
2
votes
1answer
48 views

The Fundamental Theorem of Calculus and Derivatives

How do I show this in a convincing manner? I know I need to use the Fundamental Theorem of Calculus, but I find it difficult to show any steps in between, as it appears obvious?
0
votes
1answer
130 views

Approximation of $x!$ - Proof needed

By drawing a graph of the geometric derivative of $x!$, $e^{\left(\frac{\text{d}ln(x!)}{\text{d}x}\right)}$, i guessed that $e^{\left(\frac{\text{d}ln(x!)}{\text{d}x}\right)}\sim_{+\infty}(x+1/2)$. ...
1
vote
2answers
74 views

Can we prove the formula for surface of revolution?

This is math. We like to prove things. However, proofs are rigorous processes (for a good reason) and are more than just "that idea looks like it could make sense". I've seen proofs for many ...
0
votes
2answers
42 views

Integrable functions and absolute values

I have qutoted that the absolute value of an integral is less than or equal to the integral of an absolute value of a function. I have also said $|-g(x)| \le g(x) \le |g(x)|$ implies the integral ...
12
votes
1answer
137 views

Why is an equation necessarily dimensionally correct?

I have just read a fascinating proof of the value of the integral $$ \int_{-\infty}^\infty e^{-ax^2} dx, $$ which proceeds by dimensional analysis, as follows: we know that we can write $$ ...
1
vote
0answers
44 views

expected value with integration

For the exponential distribution, $f(x)=(1/\theta) e^{-x/\theta}$ for $x>0,$ and $f(x)=0$ for $x \leq0$ $(i)$ Determine the exact value for the probability $P(0<X<3\theta).$ I need help ...
1
vote
0answers
31 views

Lifetime of pdf disk

The pdf for the lifetime X, in years, of a Superstuff disk drive is given as follows: $f(x) = \begin{cases} 2/x^2 & \text{for } x\geq2\text{ } \\ 0 & \text{elsewhere} \end{cases}$. ...
3
votes
1answer
96 views

Complex integral $ \int_{\partial D_R} \frac{\exp\bigr( \pi i (z - 1/2)^2 \bigl)}{1 - \exp(-2\pi i z)} \mathrm{d}z $

I have been working on the following problem from Gamelin VII.1 problem 6. Consider the integral $$ J = \int_{\partial D_R} \frac{\exp\bigr( \pi i (z - 1/2)^2 \bigl)}{1 - \exp(-2\pi i z)} ...
1
vote
1answer
59 views

Proof of PDF Integrals

Hi guys my professor gave us some sample proofs to try at home and I was having trouble with 4 of them. I figured out how to do part (a) by using polar coordinates but cannot wrap my head around the ...
1
vote
3answers
63 views

How can you prove this equality?

I am trying to figure out the following equality, but cannot seem to get anywhere. I tried integrating by parts, but that blew up when I set u = (log x)^n and tried to take log (0). I also tried ...
1
vote
2answers
134 views

How many digits do we need for a proof ??

In the question: Integral $\int_1^\infty\frac{\operatorname{arccot}\left(1+\frac{2\pi}{\operatorname{arcoth}x-\operatorname{arccsc}x}\right)}{\sqrt{x^2-1}}dx$, the value of that integral was ...
16
votes
2answers
867 views

Integral $\int_1^\infty\dfrac{dx}{1+2^x+3^x}$

Can the integral $$\int_1^\infty\dfrac{dx}{1+2^x+3^x}$$ be given in closed form? This question arises naturally when I considered doing integrals. What makes an integral hard? Well, the integrand, of ...
1
vote
2answers
111 views

Introduction to Analysis: The Riemann Integral

The following is a problem from Arthur Mattuck's book, "Introduction to Analysis." Page 265. Assume $f(x)$ integrable on $I$. Prove $F(x) = \int_a^x f(t)\,dt$ is continuous on $I$ How would I ...
2
votes
1answer
56 views

Proof that the Riemann-Integral satisfies $\int_A \lambda f = \lambda \int_A f$

Suppose $A\subset\mathbb{R}^n$ is a closed rectangle and $f:A\to \mathbb{R}$ is Riemann-Integrable on $A$. I want to show that $\lambda f$ is integrable and that $$\int_A \lambda f =\lambda\int_Af $$ ...
1
vote
2answers
280 views

Find the integration of $\sec(x)$ and prove it

My hw told me to prove the integral of $\sin(x)$, $\cos(x)$, $\tan(x)$, but when I get to $\sec(x)$ I'm stuck. I can find a way to prove it. Please help on explaining the integral of $\sec(x)$. ...
2
votes
0answers
217 views

Cauchy's theorem for integral homotopic closed curve in $G\subset\mathbb{C}^n$.

Recall Cauchy's theorem (third version in the Conway's book "Function of one complex variable", thm 6.7. page 90 in the second edition): Let $f$ be an analytic function on $F\subset\mathbb{C}$ and ...
2
votes
0answers
208 views

Proof of a method to find the points of maximum slope

According to method described in a paper [1] if we want to find points of maximum slope in a signal $f(t)$, then one has to do following Convolve $f(t)$ with $g(t)$ where $g(t)=-cos(\omega ...
5
votes
5answers
164 views

interesting Integral , alternative solution.

Show the following relation: $$\int_{0}^{\infty} \frac{x^{29}}{(5x^2+49)^{17}} \,\mathrm dx = \frac{14!}{2\cdot 49^2 \cdot 5^{15 }\cdot 16!}.$$ I came across this intgeral on a physics forum and ...
14
votes
4answers
1k views

Prove $\int_0^{\infty } \frac{1}{\sqrt{6 x^3+6 x+9}} \, dx=\int_0^{\infty } \frac{1}{\sqrt{9 x^3+4 x+4}} \, dx$

Prove that: $(1)$$$\int_0^{\infty } \frac{1}{\sqrt{6 x^3+6 x+9}} \ dx=\int_0^{\infty } \frac{1}{\sqrt{9 x^3+4 x+4}} \ dx$$ $(2)$$$\int_0^{\infty } \frac{1}{\sqrt{8 x^3+x+7}} \ dx>1$$ What I do for ...
2
votes
0answers
225 views

Bounds for the exponential integral

In Abramowitz and Stegun: Handbook of Mathematical Functions (on page 229, property 5.1.20) it is found that $$ \frac{1}{2} \log \left(1 + \frac{2}{x} \right) < \exp(x) E_1(x) < \log \left(1 + ...
3
votes
1answer
118 views

Show $g(\mathbf{x}) \leq h(\mathbf{x})$ implies $\int g(\mathbf{x})\mathrm{d}\mathbf{x} \leq \int h(\mathbf{x})\mathrm{d}\mathbf{x}$

Suppose I have $g$ and $h$ from $\mathbb{R}^p\to\mathbb{R}$ such that for all $\mathbf{x}$, $g(\mathbf{x}) \leq h(\mathbf{x})$. I want to prove that the integral over all $\mathbb{R}^p$ of $g$ is less ...
2
votes
1answer
268 views

proof that a function is integrable on a interval $[a,b]$

a) Divide a interval $[a,b]$ into $n$ equal subintervals. here I'm thinking $P_{n} =(x_0,x_1,x_2,x_3,x_{n-1}, x_n)$ where $a = x_0 < x_1 < x_2 < x_3 <\dots< x_{n-1} < x_n = b$ ...
5
votes
3answers
204 views

Why is $\int\limits_{1}^{n} \log x \,dx \le \sum\limits_{x = 1}^{n}\log x$?

It has been a long time since I studied integrals, so this question may sound stupid. I was going through this wiki page, and came across the following inequality: $$\int_{1}^{n} \log x \,dx \le ...
3
votes
2answers
692 views

integral property proof

I am having trouble with the following proof: Prove that if $f$ is differentiable on a closed interval $[a, b]$ then for every continuous function $g$ with the property $\int\limits_a^bf(x)g(x)dx = ...
2
votes
3answers
320 views

The Lebesgue and Riemann integrals of an increasing function over $[a,b]$ are the same.

Want to show: $f$ is Lebesgue integrable and the value of the Lebesgue integral is the same is the Riemann integral. (We're not supposing that these two are equal when the Riemann integral exists). I ...
1
vote
1answer
20 views

Simplification of Equation Involving Second Partials

I was reading this article and I'm trying to follow this author's proof. The author jumps from $$\psi_1(x)\frac{\partial^2\psi_2(x)}{\partial x^2}-\psi_2(x)\frac{\partial^2\psi_1(x)}{\partial ...
0
votes
2answers
241 views

Proof strategy for Pointwise converging sequence of Riemann integrable functions to not uniformaly converge

I am wondering of a proof strategy to show. That a sequence of Riemann integrable functions which converges point wise to a function may not actually uniformly converge to it. If it makes the argument ...
3
votes
1answer
358 views

Integration Problem Proof ($\sin x$)

Problem: Integration of $\displaystyle\int_{-1}^1 {\sin x\over 1+x^2} \; dx = 0 $ (according to WolframAlpha Definite Integral Calculator) But I don't understand how. I tried to prove using ...
4
votes
1answer
1k views

Integrating $\int \sin^n{x} \ dx$

I am working on trying to solve this problem: Prove: $\int \sin^n{x} \ dx = -\frac{1}{n} \cos{x} \cdot x \ \sin^{n - 1}{x} + \frac{n - 1}{n} \int \sin^{n - 2}{x} \ dx$ Here are the steps that I ...
0
votes
0answers
74 views

Representation of an equality

I know that I keep asking the similar problems with a little modification but it is really important to me to make sure that I am at the right track. This is my previous question link. Since we can ...
0
votes
1answer
71 views

How to prove this zeta function?

Prove that $\sum_{n=2}^{\infty} \frac{z^{n-1}}{\alpha(n-1)+1}$ is equivalent to $\frac{1}{\alpha} \displaystyle \int_{0}^{1}{ \frac{z t^{\frac{1}{\alpha}}}{1-tz}} dt$?
51
votes
7answers
7k views

Lesser-known integration tricks

I am currently studying for the GRE math subject test, which heavily tests calculus. I've reviewed most of the basic calculus techniques (integration by parts, trig substitutions, etc.) I am now ...