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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3answers
71 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
votes
1answer
52 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
vote
1answer
102 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} + ...
-3
votes
0answers
27 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
votes
1answer
60 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
vote
1answer
42 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
votes
2answers
39 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
vote
1answer
21 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
22 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
42 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
18 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
13 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 ...
8
votes
3answers
118 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
vote
3answers
50 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
46 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
vote
1answer
43 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
77 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
30 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 ...
1
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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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votes
1answer
33 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
56 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
vote
1answer
50 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
56 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
20 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
28 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
74 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
vote
0answers
22 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
30 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
45 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 ...
5
votes
1answer
77 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
32 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 ...
0
votes
2answers
57 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 ...
0
votes
0answers
14 views

Looking for guidance on a Complex Integral

I have stumbled across the following integral $$\int_{0}^{\infty } t^{\frac{1-2 H}{H}} \exp \left(-\frac{1}{2} (1-H)^{2 (H-1)} H^{-2 H} \mu ^{2 H} t^{2 (1-H)}+i t x\right) \, dt$$ where $0< H < ...
5
votes
3answers
91 views

Putnam definite integral evaluation $\int_0^{\pi/2}\frac{x\sin x\cos x}{\sin^4 x+\cos^4 x}dx$

Evaluate $$\int_0^{\pi/2}\frac{x\sin x\cos x}{\sin^4 x+\cos^4 x}dx$$ Source : Putnam By the property $\displaystyle \int_0^af(x)\,dx=\int_0^af(a-x)\,dx$: $$=\int_0^{\pi/2}\frac{(\pi/2-x)\sin ...
0
votes
0answers
34 views

Finding $\int { \exp\left(\frac { -\gamma { r }^{ 2 } }{ 2 } \right) } \exp\left(-ik \cdot r\right) { d }^{ 3 }r$ [duplicate]

Please, help me to figure out the final answer of this integral and if it is a famous form of integral, please describe it. $\displaystyle\int { \exp\left(\frac { -\gamma { r }^{ 2 } }{ 2 } \right) ...
2
votes
2answers
36 views

Jacobi Elliptic Functions Special Case

I have spent some time analysing the pendulum problem, and hence the Jacobi elliptic functions recently, and have come across what seems to me to be a slight inconsitency. I define my $am(t|k)$ as the ...
2
votes
1answer
20 views

$ \int_{ABC} f = \int_{CDA} f $

Problem from this year's MIT-PRIMES application: Let $f$ be a continuous function on the plane. In any rectangle $ABCD$ so that $AB$ is parallel to the $x$-axis and $B$ has a greater ...
3
votes
3answers
93 views

$ \lim_{n \to \infty} \int_0^{\frac{\pi}{2}} \sum_{k=1}^{n} \left( \frac {\sin kx}{k} \right)^2 \, \mathrm{d}x $

Here is a problem in calculus shared by a friend. Compute $$ \lim_{n \to \infty} \displaystyle\int_{0}^{\frac{\pi}{2}} \displaystyle\sum_{k=1}^{n} \left( \frac {\sin kx}{k} \right)^2 \, \mathrm{d}x. ...
0
votes
0answers
21 views

vector variable of integration for surface integral

$$ c(\mathbf{x}, t) = \displaystyle\int_{S} \Psi (\mathbf{x}, \mathbf{p}, t) \, \mathrm{d} \mathbf{p}. $$ Hi, I seem to have totally forgotten how we worked out an integral like above, i.e. how could ...
1
vote
0answers
22 views

Prove that a function lies in $L^1$ and in $W^{(1,1)}$ for some parameter

I want to do the following tasks Let $G:=B_1(0)\subset \mathbb R^2$ be the open ball around $0$ with radius 2 in the norm $||\cdot||$ and $u_{\rho}(x)=||x||^{\rho}_2$, $x\in G$. Show the following ...
0
votes
4answers
51 views

Power series for $f(x) = \frac{4}{x+2}$

Find the power series $f(x) = 4/(x+2)$ We know the geometric series: $$\sum_{n=1}^{\infty} x^{n-1} = \frac{1}{1-x}$$ $(x+2) = 1 - (-x - 1)$ So: $$\sum_{n=1}^{\infty} (-1)^{n-1}\cdot(x + 1)^{n-1} ...
1
vote
1answer
48 views

Evaluation of an integral when the exponent is a real number

Is there any general method for finding the following integral, $$\displaystyle\int\tan^x\theta\ d\theta$$ Where $x\in \mathbb{R}$. For $x\in \mathbb{N}\cup \{0\}$ we can easily find a recursion. But ...
1
vote
0answers
20 views

Surface area of cylindrical surface using double integrals

Please help lead me in the right direction for this question, I'll give a description of my progress so far. My understanding is that the formula for the surface area is given by this equation: ...
2
votes
0answers
24 views

Solving String Vibration Using Integral Transform

$$U_{tt} - c^2 U_{xx}= -g$$ where BC: $U_{x}(0,t)=a\sin(ωt)$ IC: $U(x,0)=0$, $U_{t}(x,0)=0$ where $c, g, A$ and $ω$ are positive constants Normally I wouldn't post for help here but I am ...