Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of ...

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0
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3answers
44 views

How would I integrate $(x^2+1)^{\frac{3}{2}}$?

I need to find $\int(x^2+1)^{\frac{3}{2}}dx.$ I started by trying to split it into $\int (x^2+1)(x^2+1)^{\frac{1}{2}}dx$ and then integrating by parts but that didn't seem to be working out. Is there ...
0
votes
0answers
20 views

Help with integrals

$$G(u,v)=(u^2-v^2,2uv)\;\;\;A=\{0 \le u \le 2,0 \le v \le 2 \}$$ What is the area of G(A)? $$f(x,y)=\frac{1}{\sqrt{x^2+y^2}}$$ What is $\iint_{G(A)}f(X)dX$? Id like to show some work but really ...
5
votes
1answer
43 views

Non-integrable function that has an antiderivative

The wikipedia article on antiderivatives states: Non-continuous functions can have antiderivatives. [...] In some cases, the antiderivatives of such pathological functions may be found by ...
0
votes
1answer
14 views

solving multi-uniform distribution intergral

I have function as following $$\int ^{\bar x} _a (y+a-b+c)f(y+a-b,y)\,dy$$ where $x\in[\underline x, \bar x]$ and $y\in [\underline y, \bar y]$ and $\bar x =\bar y + \epsilon $ and $\underline ...
3
votes
3answers
56 views

Evaluate $\int\frac{x^2}{\sqrt{1+x+x^2}}\,dx$

The task is to evaluate $$\int\frac{x^2}{\sqrt{1+x+x^2}}\,dx$$ My best approach has been substitution of $u=x+\frac{1}{2}$, and from there onto (some terrible) trig sub - finally arriving at a messy ...
1
vote
1answer
26 views

$\int\limits_{\gamma} \frac{z}{(z-1)(z-2)}$, $\gamma(\theta) = re^{i\theta}$, $2 < r < \infty$

For $0 < r < 2$, we can use Cauchy's integral formula and choose our holomorphic function to be $f(z) = \frac{z}{z - 2}$ since $z = 1$ is the only pole, but if $r > 2$, then both poles $z = ...
3
votes
3answers
38 views

Using substitution to make an integral trivial

Consider the integral $$ \int_0^\pi \frac {\cos(\theta)} {f(\sin(\theta))}d\theta $$ Assume that $f(\sin(\theta))$ is nonzero on $[0,\pi]$. Can we use the substitution $u=\sin(\theta)$ to make the ...
1
vote
1answer
38 views

Integral using trigonometric substitution

I'd like to ask for feedback on my calculation for this integral: $$\int{\frac{dx}{2-\cos{x}}}$$ Using half-angle substitution: $$t = \tan{\frac{x}{2}}$$ $$\cos{x} = \frac{1-t^2}{1+t^2}$$ $$dx = ...
0
votes
2answers
46 views

$\int^\infty_0 e^{-\alpha x}\sin(\beta x)\,dx = \frac{B}{\alpha^2+\beta^2}$ Laplace

$$ \int^\infty_0 \! e^{-\alpha x} \sin(\beta x)\,dx = \frac{\beta}{\alpha^2+\beta^2} $$ Can someone start this for me? I don't know where to begin.
1
vote
1answer
63 views

derivative of the integral

I am working on a few problems, just need some help to see if I'm working them correctly, $$(1)\;\;\;\;\;g(x)=\int_0^x(x-u)e^{u^2}du$$ find $g'(x),g''(x)$ $$(2)\;\;\;\;\;\psi(x,y)=\int_1^xe^{ty}dt$$ ...
-2
votes
0answers
9 views

Work-integration problem for a parabolic trough [on hold]

A trough is 4 feet long and 1 foot high. The vertical cross-section of the trough parallel to an end is shaped like the graph of y=x4 from x=−1 to x=1. The trough is full of water. Find the amount of ...
2
votes
3answers
72 views

Computing $\int_3^5 \frac{x^2\,dx}{\sqrt{(x-3)(5-x)}}$

$$ \int_3^5 \frac{x^2\,dx}{\sqrt{(x-3)(5-x)}} $$ how? $x^2/\sqrt{8x-x^2-15}$ and what to do then?
2
votes
3answers
129 views

Evaluating $\int\frac{x^4+1}{x^6+1}dx$

I have problem with this integral: $$\int\dfrac{x^4+1}{x^6+1}dx$$ I guess it is easy, but I was trying for quite a long time and the only thing I got is headache. Thanks for help
2
votes
1answer
26 views

Cartesian into polar integral.

I have set up an double integral to prove gauss theorem in physics for a gaussian surface of cube of edge $a$ which is as follow. I supposed that mid point of cube is at origin and a charge is placed ...
1
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1answer
21 views

Integral of polynomial using substitution

I have an integral problem that I'm working on, it's a polynomial which I imagine either can't be factored or needs to be completed, and then substituted using a trig identity: ...
1
vote
1answer
19 views

$\int_{\Omega\setminus A_n}f\;d\mu\to\int_\Omega f\;d\mu$ for all measurable $A_n\downarrow\emptyset$

Let $(\Omega,\mathcal{A},\mu)$ be a measure space $(A_n)_{n\in\mathbb{N}}\subseteq\mathcal{A}$ such that $A_n\downarrow\emptyset$, i.e. $A_n\supseteq A_{n+1}$ and ...
1
vote
2answers
96 views

Evaluate the integral $\int \sin(x)\cos(3x^2)dx$

I am looking for a solution for the following integral problem. $$\int \sin(x)\cos(3x^2)dx$$ Passed over these integral things long time ago. I cannot see how to go for a solution.
4
votes
2answers
59 views

This one wierd trick integrates fractals. But does it deliver the correct results?

It occurs to me that people most likely already know how to explicitly integrate over fractals, but my method seems to vastly simplify the process (So even a comparative layman like me can do it). ...
0
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0answers
14 views

Derivatives of semi-elasticities

I want to find the sign of $\partial_z (\partial_z \log F(T1) - \partial_z \log F(T2) )$ where $T1<T2$ and $F(T) = \int_0^T e^{g(t) + z\, h(t)} dt$ All we know is that $h(t)>0\forall t$ ...
0
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0answers
8 views

Accurate numerical integration for “data times an analytical function”

The Question is as follows: I have an algorithm/data that provides me the value of a function $f(x,y,z)$ on the points of a grid. On the other hand I have an analytical function ...
2
votes
3answers
45 views

Find the value of the integral $\int_0 ^\sqrt2 \sqrt{4-x^2} \ dx$

Find the value of the integral $\int_0 ^\sqrt2 \sqrt{4-x^2} \ dx$ I was unsure how to do this question so I looked at the mark scheme, and it said use $x=2\sin\theta$ and so $dx=2\cos\theta \ ...
1
vote
2answers
26 views

Evaluating integrals with trigonometric function

Now I have to evaluate the integrals $$ \int_{0}^{\pi /2} \sin^2 t \cos t dt $$ $$ \int_{0}^{\pi /2} \cos t \sin^2 t dt $$ $$ \int_{0}^{\pi /2} \tan^2 t dt $$ For the first two integrals, I could ...
0
votes
1answer
20 views

Undefined Subintervals - Riemann Integrals

I searched through stackexchange and multiple other PDFs but couldn't find an answer I'm curious to know when talking about Riemann Integrals with respect to functions that are bounded on closed ...
-4
votes
0answers
26 views

Integration everywhere [duplicate]

Find this $\displaystyle\int\frac{e^ x}{x}\;dx$
2
votes
4answers
71 views

How to find $\int \frac{\ln(x)}{x^2}dx$

I need to find $$\int \frac{\ln(x)}{x^2}dx.$$ I have tried substitution with $u=\ln(x)$, then $du = 1/x dx$, but this only takes care of one of the $x$ on the bottom: $$ \int \frac{u}{x} du. $$ I ...
-1
votes
0answers
40 views

Help with setting up this double integral [on hold]

"By evaluating an appropriate double integral, find the volume of the wedge lying between the planes $z=px$ and $z=qx$ (with $p>q>0$) and the cylinder $x^2+y^2 =2ax$ (where $a > 0$)." I'm ...
0
votes
0answers
16 views

How do I find the mass of a circular disk?

In this question I have been given the specific weight of the disk as 15kN/m^3 and have the radius of the disk as 0.25m. how do I find the mass of the disk? (To then go on and find the moment of ...
2
votes
2answers
39 views

Integral by using substitution (How to proceed?)

Using the substitution $x=a\sin\theta$, or otherwise, find $\int\frac{1}{x^2\sqrt{a^2-x^2}}dx$. My attempt, $x=a\sin\theta$ $dx=a\cos (\theta)d\theta$. Then $\sqrt{a^2-x^2}=\sqrt{a^2-a^2\sin ...
0
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0answers
38 views

Integral of the reciprocal of the natural logarithm [on hold]

What is the value of this integral $$\displaystyle\int\frac{1}{\ln(x)}\;dx$$
1
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0answers
11 views

Euler method global truncation error and conservation of orbital energy.

I've been given a simplified model of a small mass orbiting a much larger one, which I need to solve using Euler's method. I've reduced the equation to two (or four, really) coupled first order ODEs: ...
0
votes
0answers
8 views

if $F(s_{0})$ for some $s_{o}$exists then it exists for all $s>s_{o}$

if laplace transform $F(s_{0})$ for some $s_{o}$exists then it exists for all $s>s_{o}$. i need to prove this . now, ...
1
vote
1answer
14 views

Determining Line Integrals from a Graph and Vector Field (Image Included)

Consider the vector field: $$F=\left(\frac{2xy-2xy^2}{\left(1+x^2\right)^2}+\frac{8}{13}\right)i+\left(\frac{2y-1}{1+x^2}+2y\right)j$$ Determine $$\int_C F\cdot dr$$ where $C$ is the path ...
0
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0answers
35 views

Diffrental equation solution [on hold]

How can I solve this equation? $$\frac{\partial f}{\partial x} =\frac{a-x}{y} \frac{\partial f}{\partial y}$$ where $a$ is a constant. So what is $f$?
0
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1answer
49 views

Integral of a total derivative

I have seen the "total differential" $$ d \ln A = -d \ln B/c $$ Representing how infinitesimal changes in $A$ are related to infinitesimal changes in $B$. Someone then took the integral of this ...
3
votes
1answer
36 views

Volume of a Solid of Revolution Rotated Around the Y-Axis

Sorry to post an obvious homework question here, but my daughter's calculus teacher isn't much on "teaching" and left a problem like this one out of the notes. I can't find much on the internet to ...
1
vote
2answers
48 views

Antiderivative of $ (x^2 + c)^{-3/2} $ [on hold]

What method should be used to determine the antiderivative of this expression? Edit: I have $ c > 0 $ in the problem I'm working on.
-1
votes
2answers
26 views

Integrating to find deceleration, and finding ball height? [on hold]

1) A ball is thrown straight up from a height of 8 ft with an initial velocity of 40 ft/sec. How high will the ball go? (Take g = 32 ft/sec2.) How would I do this? Wouldn't I need to find a velocity ...
3
votes
1answer
43 views

Given $f\in L^1(\mathbb{R})$ with $\|f\|_1<\infty$ and $g_n=\sqrt{n/2\pi}e^{-nx^2/2},f_n=g_n\ast f$, show that $\lim\|f_n-f\|_1=0$

Given $f$ a Lebesgue integrable function on $\mathbb{R}$ with finite $L^1$-norm, I am asked to show that $\lim_{n\to\infty} \|f_n - f\|_1 = 0$, where $f_n = f \ast g_n$ and $g_n = ...
2
votes
1answer
68 views

Showing that $\int fg\le \int g$ implies $f=0$ a.e.

Take $0<p<1$. If $f$ is locally integrable over on $\mathbb{R}$ and $$\Bigg\vert \int fg\Bigg\vert\le \Vert g\Vert_p\tag 1$$ for every $g$ continuous on a set of compact support, then $f=0$ a.e. ...
1
vote
0answers
15 views

The $L^p$ convergence rate of the tail of the series $\sum_{n=1}^{\infty}\min\{1,2^n |x|^{-1} \}2^{-na}$

This a follow-up to the question: Convergence Rate of the Tail of the Series $m^{a}\sum_{n=1}^{\infty}\min\{1,2^n m^{-1} \}2^{-ja}$ When $a > 0$, we have $$ \sum_{n=1}^{\infty}\min\{1,2^n |x|^{-1} ...
0
votes
1answer
13 views

Lower semicontinuous non-negative function on a locally compact Hausdroff space with a countable base

An extended real number is an element of $\mathbb R \cup \{-\infty, +\infty\}$. Let $X$ be a locally compact Hausdorff space with a countable base. An extended real valued function $f$ on $X$ is ...
0
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0answers
28 views

Any hint for solving this Poisson's integral? [on hold]

I tried various approach without success in solving this integral: $\frac{1}{2\sqrt{\pi t}}\int_{\mathbb{R}} e^{\frac{-(x-y)^2}{4t}}\phi (y) dy$ which is the solution to the heat equation. I only have ...
2
votes
1answer
40 views

Integration over ellipse

$A=\{(x,y)\in \Bbb R^2\mid \frac{x^2}{a^2}+\frac {y^2}{b^2}=1\}$. Find $\int_A (\cos x)y\,dx+(x+\sin x)\,dy$. Can someone please please give a methodological answer? Thanks a lot!
0
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0answers
19 views

A question of multi-dimensional integral

Consider the function $$\Omega(N,E)=\int dE_1 \int dE_2 \cdots \int dE_N \Omega_1(E_1)\Omega_2(E_2) \cdots \Omega_N(E_N)\delta(E-E_1-E_2\cdots -E_N)$$ Is there a sufficiently condition on the ...
2
votes
1answer
74 views

Bonus integration problem we got at class: Integrate $\frac {x \sin x}{1+\cos^2x}$ between $0$ and $\pi$ [duplicate]

Bonus integration problem we got at class: Integrate $\frac {x \sin x}{1+\cos^2x}$ between $0$ and $\pi$ So the lecturer gave this problem. I tried this really hard but couldn't succeed. It ...
0
votes
3answers
71 views

Integrate $\frac{1}{1+\cos^2x}$. Probably need using some trigonometric identity I don't know

Integrate $\frac{1}{1+\cos^2x}$ I probably need using some trigonometric identity I don't know. I tried all methods I'm familiar with. Any assistance will be great. Thank you!
1
vote
2answers
49 views

How does the first fundamental theorem of calculus guarantee the existence of antiderivatives of functions?

First fundamental theorem of calculus: $$g(x) = \int_a^xf(t)dt$$ then $$g'(x) = f(x)$$ But how does this guarantee the existence of antiderivatives of functions? Tutorials always state it does, but ...
3
votes
3answers
62 views

Integral of trig fraction using substitution?

I'm chewing on an integral problem and don't have a clue where to begin. If someone could assist by suggesting a good starting point, I'd really appreciate it! Not asking for anyone to solve the ...
0
votes
2answers
30 views

Evaluating a complex integral (Hints please)

I am supposed to be able to show that, given $f(z)=\frac{1}{\pi}\int_0^1r\int_{-\pi}^\pi\frac{d\theta}{re^{i\theta}+z}dr$ then $f(z)=\overline{z}$ for $|z|<1$ and $f(z)=1/z$ if $|z|\geq1$. (This ...
1
vote
0answers
38 views

Integration of a function of two variables

How can we check the integrability of $f$ defined on $[0,1] \times [0,1]$ as $f(x,y)=$\begin{cases} 0 & x=\frac{1}{2},y \in \mathbb Q \\ 1 & x=\frac{1}{2},y \in \mathbb Q^c \\ ...