All aspects of integration, including the definition of the integral and computing different types of integrals. For questions solely about the properties of integrals, don't use this tag alone! Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another ...

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1answer
17 views

Bound on and integral

If $\alpha \in \Bbb R$, how can I show $$\int_{-M}^M \frac{1}{\sqrt{|x-\alpha|}} \, dx \le 4 \sqrt{M}$$ For $M>0$, Rewriting the integral gives $$\int_{-M+\alpha}^{M + \alpha} ...
1
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0answers
15 views

product of barycentric coordinates over a simplex

Show that: $$\int_K \eta_0^{\alpha_0}(x)\cdots\eta_n(x)^{\alpha_n}dx=\frac{{\alpha_0}!\cdots{\alpha_n}!n!|K|}{(\alpha_0+\cdots+\alpha_n+n)!}$$ where $\eta_i$ barycentric coordinates and $K$ is a ...
2
votes
3answers
21 views

Determine monotone intervals of a function

Let $$ f(x) = \int_1^{x^2} (x^2 - t) e^{-t^2}dt. $$ We need to determine monotone intervals of $f(x)$. I tried to differentiate $f(x)$ as follows. $$ f'(x) = \left(x^2 \int_1^{x^2} e^{-t^2}dt \right)' ...
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0answers
16 views

Finite integral involving branch cut. Basic Argument Question

I am reading this Wikipedia article on examples of contour integrals using complex analysis (http://en.wikipedia.org/wiki/Methods_of_contour_integration). In particular, I am looking at Example (VI), ...
2
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8answers
106 views

How to show that $f(x) = 0$ if $\int_a^bf(x)\,\text{d}x=0$ for all $a,b\in\mathbb{R}$?

I found this problem on the web: Let $f(x)$ be a real-valued, continuous function with the property that $$\int_a^bf(x)\,\text{d}x=0$$for all real numbers $a,b$. Prove that $f$ is identically $0$. ...
3
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7answers
86 views

Proving that $\int_0 ^1 \frac{\text{d}s}{\sqrt{1-s^2}}$ converges with no trig functions

Let $$\int_0 ^1 \frac{\text{d}s}{\sqrt{1-s^2}}$$ How to show that it converges with no use of trigonometric functions? (trivially, it is the anti-derivative of $\sin ^{-1}$ and therfore can be ...
1
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0answers
10 views

How to calculate convolution of function defining a measure

Given the function $F(t)=2-2e^{-t}$ defining a measure on $(\mathbb{R}_+,\mathfrak{B}(\mathbb{R}_+))$ and I want to calculate the convolution of this function with itself. I tried to do that by using ...
1
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1answer
29 views

How to compute $\int^{1}_{-1}f(x)dx$?

I need to compute $\int^{1}_{-1}f(x)dx$, where $$f(x)=\{^{x,\ x \leq 0}_{x^2,\ x \geq 0}$$ I know that $\int^{1}_{-1}xdx=0$ & $\int^{1}_{-1}x^2dx=\frac{2}{3}$ Is it right?
12
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0answers
122 views

Evaluate $\int_0^{\pi/2} \frac{\ln\left(e^{2x} + 1\right)}{1 + \sin2x}\mathrm dx$

Here's my Xmas gift to all of you! I just encountered a very tough integral. $$\int_{0}^{\pi/2} \frac{\ln\left(e^{2x} + 1\right)}{1 + \sin2x}\mathrm dx$$ I have tried for a few hours. This task is ...
8
votes
3answers
131 views

How to solve the differential equation $(2x^3y)\:\text{dy}+(1-y^2)(x^2y^2+y^2-1)\:\text{dx}=0$?

Solve $$(2x^3y)\:\text{dy}+(1-y^2)(x^2y^2+y^2-1)\:\text{dx}=0$$ I tried the substitution $y^2=t$ ; $2y\:\text{dy}=\text{dt}$ to get $$(x^3)\:\text{dt}+(1-t)[(x^2+1)t-1]\:\text{dx}=0$$ ...
4
votes
2answers
93 views

How to integrate a fraction of the type $\frac{1}{(ax+b)^c(dx+e)^f}$?

I'm working on obtaining chemical reactions' speed, and this is one of the problems I met with. $$ \int \frac{1}{(ax+b)^c(dx+e)^f}dx $$ Can this equation could be solved? If possible, please show ...
0
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1answer
26 views

Is this enough to demonstrate divergence of an improper integral?

The integral in question is $$\int_0^\infty (f(x)-a)^2dx$$ Where f(x) is some continuous function and a is some constant. When we expand the integrand,we end up with an $a^2$ term. We can then ...
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3answers
95 views

Evaluation of the integral $\int 3x \cos x^2 \, dx$

I want to solve this: $$\int 3x \cos x^2 \, dx$$ I get this answer: $$ \frac{\sin 2x}{2}+\frac{\cos 2x}{4}+C $$ but the answer should be: $$ \frac{3 \sin x^2}{2}+C $$ Am I doing anything wrong ...
0
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1answer
64 views

How to approach, substitution - definite integral

So I have this problem $${\int^{\pi/2}_0} {{\cos\theta \sin\theta}\over \sqrt{\cos^{2}\theta +8}}d\theta $$ and I'm not sure if this is the right direction to begin. If I have $u = \cos\theta$ ...
1
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1answer
107 views

A question from my final exam

Today I had the final exam of the lesson Mathematics I. There was a question that I want to know if I solved correctly the following limit. $$\lim\limits_{n\to\infty}\left(\frac{n}{0^2+n^2} + ...
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0answers
42 views

Show an integral is bounded under a parameter $\alpha \in (0,1)$ [on hold]

Set $f:[0,1] \to \mathbb{R}$ a continuous function that $f(0)>0$. Show that limit given by: $$\lim_{h \to 0+} \int_{h}^1 \frac{f(x)}{x^{\alpha}}dx$$ exist and is finite for all $\alpha \in ...
2
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1answer
74 views

Evaluating $\int_0^{2} \frac{dx}{\sqrt[3]{2x^2-x^3}}$

How to calculate this integral? $$\int_0^{2} \frac{dx}{\sqrt[3]{2x^2-x^3}}$$ I suppose that it should be parted like this: $$\int_0^{1} \frac{dx}{\sqrt[3]{2x^2-x^3}} + \int_1^{2} ...
1
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1answer
48 views

Generating functions for $\log^3(1-x)$ of $\log^3(x)$

I am trying to find generating functions which will give me a power logarithm. I am trying to find generating sums in the form $$\sum_{n=1}^{\infty} a_n\,x^n = -\frac{\log^2(1-x)}{1-x}$$ or ...
0
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2answers
42 views

Taylor expansion of the Error function

The error function $\operatorname{erf}(z)$ is defined by the integral $$ \operatorname{erf}(z)=\frac{2}{\sqrt{\pi}} \int_0^z e^{-t^2}\,dt,\quad t\in\mathbb R$$ Find the Taylor expansion of ...
1
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1answer
29 views

Composition of a Dirac delta and a function in higher dimensions

Coming from a physics background, I was taught the formula for the composition of a Dirac delta and a function. Indeed, if we consider a nice function $ f : \mathbb{R} \to \mathbb{R} $, one can write ...
2
votes
1answer
43 views

Integral over a simplex

Let $C_k$ be the $k$-simplex. I know that $$\int_{C_k} \prod_{i=1}^k x_i^{\alpha_i-1} dx_i = \frac{\prod_{i=1}^k \Gamma(\alpha_i)}{\Gamma\left(\sum_{i=1}^k \alpha_i\right)} \equiv ...
4
votes
2answers
47 views

Calculating Triple Integral

I have task : find volume of body limited by surface $(\frac{x}{a})^{2/3} + (\frac{y}{b})^{2/3} + (\frac{z}{c})^{2/3}$ = 1. I know that this task is about triple integral. But i have confused by such ...
0
votes
1answer
24 views

Fourier transform, quadratic function

I'm trying to compute this convolution: $\frac{2 \alpha}{\alpha ^2 + 4 \pi ^2 x^2} * \frac{2 \beta}{\beta ^2 + 4 \pi ^2 x^2}$ I know that the Fourier transform of a convolution of two functions is ...
0
votes
0answers
15 views

Fourier transform of $be^{i k y^b}/y^{1-b}$

I'm trying to compute the Fourier transform of $$ \frac{ be^{i k y^b}}{y^{1-b}}$$, i.e. $$ F(z) = \int_{-\infty}^\infty \frac{ be^{i k y^b}}{y^{1-b}} e^{i z y}dy$$ I tried using Mathematica for ...
10
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3answers
143 views

Proving $\int_{0}^{\infty}\frac{x}{(x^2+1)(e^{2\pi x}+1)} dx=1-\frac{\gamma}{2}-\ln2$

Nowadays I encounter an integral which is difficult for me to evaluate it. Please help me to evaluate it. Thank you. $$\int_{0}^{\infty}\frac{x}{(x^2+1)(e^{2\pi x}+1)} ...
1
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3answers
52 views

Does there exist a continous function $f(t)$ on $[0,1]$ for which $\int_0^1 t^3 f(t) dt = 0$?

Does there exist a continous function $f(t)$ on $[0,1]$ for which $\int_0^1 t^3 f(t) dt = 0$? Or can you provide a proof otherwise?
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2answers
48 views

Does alternating test show divergence?

My book states the alternating tests' convergence requirements. However, my book doesnt point out, if $a_n$ fails one of the convergence requirements, is it true that is diverges? Such as the limit ...
0
votes
1answer
28 views

Two different results with contour integration

This is probably going to be a stupid question ( I don't feel great today) but I can't get around this problem. $$I = \int_\mathbb R \frac 1 {(3x-2i)^2} dx $$ I thought that using contour ...
1
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1answer
45 views

$ \int_\gamma \frac{1}{z\sin z}dz$ where $\gamma$ is the circle $|z| = 5$

My understanding is that if this integral exists in the real sense, i.e. real Riemann-wise, then I can apply the residue theorem. If not, I may use the Cauchy Principal Value, to obtain a value. To ...
2
votes
2answers
85 views

Integral $\int_{1}^{2011} \frac{\sqrt{x}}{\sqrt{2012 - x} + \sqrt{x}}dx$

Evaluate: $$\int_{1}^{2011} \frac{\sqrt{x}}{\sqrt{2012 - x} + \sqrt{x}}dx$$ Using real methods only. I am not sure what to do. I tried finding a power series, which was too ugly. I just need some ...
3
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0answers
36 views

An a.e.-defined derivative which is not Lebesgue integrable on any interval?

If the derivative $f'$ exists everywhere then it is shown here that there exist intervals on which $f'$ is Lebesgue integrable. But perhaps there is a function $f$ such that $f'$ only exists almost ...
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0answers
24 views

Example for Stieltjes Integral? [on hold]

I have problem make a example from this paper about Stieltjes Integral and you can check at stieltjes integral like this :
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1answer
37 views

Integrate $dx/(4x^2-1)^{3/2}$

I have trouble using trig sub. After I get that x = 2x+1, should I substitute back into the original problem's $4x^2$ with $(4(2x+1)^2)$?
1
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0answers
61 views

Calculating an integral with sine, cosine

I've recently calculated the Fourier transform of $\dfrac{\sin \pi ax}{\pi x}$. Now I'm trying to calculate $$\int _{\mathbb{R}} \frac{\sin ^2 \pi ax}{\pi ^2 x^3} \cos \pi bx\;\mathrm dx$$ The ...
1
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1answer
53 views

How to evaluate $\int \cot^2(x) \;\mathrm dx$?

How do you find the antiderivative of $\cot^2x$? My steps to find it First $$ \csc^2 x = \cot^2 x+ 1 $$ because of Pythagorean Identities, so $$ \cot^2 x= \csc^2 x-1$$ so $$ \int \cot^2 x\, ...
2
votes
2answers
48 views

Integrate $\int \csc^6(2x)\, dx$

I know to use the identity $1+\cot^2(2x)$. I'm not sure how to use $u$-substitution to substitute the $2x$ from the problem. I would have to use a $u$-substitution and then another $w$-substitution. ...
0
votes
1answer
78 views

Problems taking the limit in $\int_a^b f=\lim_{c\to a}\int_c^b f$ from definitions

Let $f$ be bounded on $[a,b]$ and Riemann integrable for each $c$ with $a<c<b$. I need to show that $f$ is Riemann integrable on $[a,b]$, and $\int_a^b f=\lim_{c\to a}\int_c^b f$. My ...
1
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0answers
21 views

How to compute $\int_{-1}^1 x^p (1-x^2)^{\frac{d-3}{2}} P_n^d(x) dx$

For a project I want to get a closed form solution of $$\int_{-1}^1 x^p (1-x^2)^{\frac{d-3}{2}} P_n^d(x) dx$$ Here $p \in \mathbb{N},\; d\ge3, \; d\in\mathbb{N}$ and $P_n^d$ is the associated ...
3
votes
1answer
29 views

Volume when rotated about the line $y=-1$

Find the volume when the region enclosed by $y=x^2$, $y=4$ is revolved around the line $y=-1$ My teacher has given the following answer: I assume she has done this through the method of shells, ...
1
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0answers
75 views

Integrals and f(x)dx

Suppose $$\int_0^2 f(x)\,dx=3$$ $$\int_0^5 f(x)\,dx=8$$ Compute $$\int_2^5 f(x)\, dx$$ $$\int_0^2 f(2x)\,dx$$ For the first one, I know that by subtraction $$\int_2^5 f(x)\,dx = \int_0^5 ...
1
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0answers
23 views

Applying Stokes' theorem

$C$ is the surface $z=y(e^{-x^2}-y-1/2)$ and conditions $z\geq 0$, $x \in [-1, 1]$ and $\varphi = dxdz+dydz+(e^{-x^2}-2y-1/2)xe^{xz}dxdy$ is a 2-form. I have to compute $\int_C\varphi$ using Stokes' ...
2
votes
1answer
25 views

Calculate surface integral

I need some help with the following: Given $$f(x,y,z)=\left( \frac{-x}{(x^2+y^2+z^2)^{\frac{3}{2}}}, \frac{-y}{(x^2+y^2+z^2)^{\frac32}}, \frac{-z}{(x^2+y^2+z^2)^{\frac32}} \right),$$ calculate the ...
0
votes
1answer
20 views

How to find the integration bounds when calculating area

To calculate an area between curves, I need to integrate with respect to x between the curve $y=\sqrt{2x}$, the x-axis and the line $y=\frac{4x-12}{5}$ My understanding, using google to display plot ...
0
votes
1answer
31 views

$C_c(\mathbb R^n)$ is not dense in $\mathcal L^\infty(\mathbb R ^n)$

I'm having some difficulties in manipulating the space $\mathcal L^\infty(\mathbb R ^n)$, and I want to show that $C_c(\mathbb R^n)$ is not dense in $\mathcal L^\infty(\mathbb R ^n)$, but I can't find ...
2
votes
1answer
46 views

Surface of revolution of an ellipse

I have been working on this question, but I end up getting the wrong answer overtime: The ellipse $$\frac{x^2}{a^2}+ \frac{y^2}{b^2} = 1$$ where $a>b$ is rotated about the $x$-axis to form a ...
1
vote
2answers
64 views

Questionable Power Series for $1/x$ about $x=0$

WolframAlpha states that The power series for $1/x$ about $x=0$ is: $$1/x = \sum_{n=0}^{\infty} (-1)^n(x-1)^n$$ This is supposedly incorrect, isnt it? This is showing the power series about ...
6
votes
1answer
80 views

Solving this complicated integral using the Residue Theorem

The following is an integral I am trying to evaluate $$I= \int_{-\infty}^\infty f(s) \, ds = \int_{-\infty}^\infty \frac{\frac{1}{(1- \ \ 2 \pi j s )^{m}}-1}{2\pi j s }\ e^{-2\pi j s \ \theta}\ ds ...
1
vote
2answers
35 views

Give an example of a function who is nondifferentiable on (0, 2) but has an antiderivative on (0, 2)

Originally when I was playing around with this problem, I tried to first find a function who was differentiable, but whose derivative was not differentiable at a specific point. So I figured out the ...
2
votes
1answer
19 views

$L_1$ convergence of $\frac{1}{\sqrt{x}}\sin{\left(\frac{1}{nx}\right)}$

Does the sequence $f_n=\frac{1}{\sqrt{x}}\sin{\left(\frac{1}{nx}\right)}$ on $(0,1)$ converge in $L_1$? It converges to zero pointwise and I think it converges in $L_1$ as well since ...
1
vote
2answers
61 views

How to find the derivative of $F(x)=\int_{x^2}^{4x^2} \sin \sqrt t\;\;dt$?

For a real number $t>0$, let $\sqrt t$ denote the positive square root of t. For a real number $x>0$, let $F(x)=\int_{x^2}^{4x^2} \sin \sqrt t\;\;dt$. If $F'$ is the derivative of $F$, then ...