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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2
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0answers
97 views

how to use Matlab ifft to calculate the following integral? [duplicate]

$$R(t)=\int_{-\infty}^\infty\dfrac{\omega e^{i\omega t}}{(3-\omega^2)^{2}+4\omega^2}\,d\omega$$ where t is a integer and $t>0$ I used to calculate this integral by numerical integral,but it seems ...
1
vote
1answer
60 views

Volume of a region R when revolved about the x-axis

Find the volume of the region when revolved about the $x$ axis $y= \sqrt{x-1}$, $y=2$, $y=0$, and $x=0$ Is this right? Also if you could help me with revolving this same region around $y=2$, ...
1
vote
2answers
59 views

If $f$ is continuous and $g$ is integrable on $[a,b]$, with $g(x) \ge 0$ for all $x \in [a,b]$ …

Suppose $f : [a,b] \to \mathbb{R}$ is continuous and $g \in \mathcal{R}[a,b]$ with $g(x) \ge 0$ for all $x \in [a,b]$. Show that there exists a $c \in [a,b]$ such that $$\int_a^b f(x)g(x) \, dx = ...
3
votes
2answers
54 views

How to find $\int \sec^{3} \ dx$ [duplicate]

I am stuck trying to find $$\int \sec^3{x} \ dx.$$ Here is my attempt using integration by parts: $$\int \sec^3{x} \ dx = \sec{x}\tan{x} - \int \tan^2{x}\sec{x} \ dx.$$ At this point, I am stuck. ...
1
vote
2answers
51 views

area under a curve and units

If we introduce a unit of length like meter for $x$ and integrate the function $f(x)=x^2$ from $0$ to $2m$ we get $\dfrac{8}{3} m^3$. How can this be interpreted geometrically? My initial thought was ...
1
vote
0answers
55 views

What does $dx$ mean in an indefinite integral?

$\int_0^{1} 3{x^3}\,{dx} $ $\int3{x^3}\,{dx} $ Both the definite and indefinite integrals have the same mark in their ends, ${dx}$. The latter one's ${dx}$ would mean an infinitely small width of ...
0
votes
1answer
42 views

Why do the limits of integration matter in a double integral?

Okay, I know that seems like a stupid question but I couldn't think of a better way to phrase it. I was trying to understand why iterated integrals involve "projecting" the domain onto one of the ...
4
votes
2answers
85 views

How to integrate $\ln \big( b + \sqrt{b^2 + c^2 + x^2}\,\big)$?

I am looking to demonstrate the following result. Any ideas are much appreciated. $$ \begin{align}\int \ln \left( b + \sqrt{b^2 + c^2 + x^2}\right) dx = &\;x \ln \left( b + \sqrt{b^2 +c^2 ...
4
votes
1answer
46 views

power series for $\int_0^x e^{-t^2}dt$

Use a known power series expansion to find the power series representation of the integral function $g(x) =\int_0^x e^{-t^2}dt$ centered at $a=0$ My approach Note that $g'(x) = e^{-x^2}$. ...
10
votes
3answers
248 views

How to find $\int_{-1}^1 \frac{\cos x}{a^x+1}\mathrm dx$

Evaluate $$\int_{-1}^1 \frac{\cos x}{a^x+1}\mathrm dx$$ where $a$ is a real parameter $a\geq1$. I can easily find the definite integral for $a=1$. It is $\sin(1)$. In wolframalpha.com when I put ...
3
votes
2answers
116 views

The integral of $e^{-x^2}$ [duplicate]

How can I integrate this by parts? It seems to become recursive. I'm familiar with the classical solution, and cannot use that here due to the constraints of this class. Here's the integral (to ...
2
votes
1answer
55 views

Measure Spaces: Uniform & Integral Convergence

Given a measure space $\Omega$. Consider a sequence of measurable functions $f_n$ Suppose it converges pointwise: $f_n\to f$ Can one find increasing subsets with uniform convergence: ...
5
votes
1answer
497 views

Riemann Integrability in $\Bbb R^2$

Define the General Subdivision $S$ of a rectangle $R$ in $\Bbb R^2$ as a collection $E_1,...,E_k$ of Jordan regions such that none of them has interior points in common, and: $$R \subset ...
3
votes
0answers
27 views

Finding the Area of a Torus-like surface

I'm trying to find out the Area of the following surface: Let $C$ be the curve associated to a regular, simple path $\theta:[0,l]\rightarrow \Bbb R^2 $; also assume that ...
2
votes
1answer
26 views

A question regarding Surface Integrals and Stoke's Theorem

Let $G$ be an open set in $ \Bbb R^3$ and $F:G \rightarrow \Bbb R^3-{0}$ a vectorial field of class $C^1$. Suppose that $S$ is an open set, contained in $G$, whose non-empty boundary $\delta S$, is ...
0
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0answers
44 views

Solving a fractional trigonometric integral

On an answer on another SE I wrote, I came upon the integral $$\int \frac{x}{a-b\cos\left(\frac{x}{2}\right)+c\left(\cos^2\left(\frac{x}{2}\right)+\sin^2\left(\frac{x}{2}\right)\right)}dx$$ Solving it ...
2
votes
2answers
105 views

Supremum of a sine integral

Let $M_T=\int\limits_{0}^{T}\frac{\sin(t)}{t}dt$ be a sine integral. Why is $2\displaystyle\sup_{T}M_T < \infty$?
2
votes
1answer
79 views

I can't solve this integral in Mathematica

I want to calculate the following integral in Mathematica 10 $$f(x) = ...
1
vote
0answers
32 views

Decisions on the order of integration with double integrals (when Deriving PDF via CDF) (Bank Problem)

Consider the following problem: Gandalf, Saruman and Radagast go to a bank together. There are two open counters which Gandalf and Saruman immediately go to get their service. Radagast goes to the ...
0
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0answers
37 views

Weber Transform

During my studies I meet the Weber Transform of the free space potential function, that is: $$\int _{\rho }^{\infty }\exp(-i \text{$\key} t) (Y_0(\text{$\lambda $p} \rho ) J_0(\text{$\lambda $p} ...
0
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0answers
30 views

Trig sub and Integration of Squareroot divided by polynomial squared

Question #2 What am I doing wrong? Do not give me the answer but rather a hint.
0
votes
1answer
368 views

Evaluate integral by interpreting it in terms of areas

I tried (a) and I got 5, but I am suppose to get a 4. I really need a good explanation to understand how to approach these problems. I tried searching in youtube and stuff, but it was not helpful. ...
5
votes
1answer
106 views

An advanced integral $\int_0^1 \frac{(2 e)^{-1/y} \left(2 e^{1/y}-e 2^{1/y}\right)}{1-y} \ dy$

I'd like to ask you how you would like to approach the integral below $$\int_0^1 \frac{(2 e)^{-1/y} \left(2 e^{1/y}-e 2^{1/y}\right)}{1-y} \ dy$$ and then recommend me some tools you'd employ. It's ...
0
votes
0answers
35 views

Rewrite the integral $\int_{0}^{1}\int_{\sqrt{x}}^{1}\int_{0}^{1-y}f(x,y,z)\,dz\,dy\,dx$ in the orders $dx\,dy\,dz$ and $dy\,dz\,dx$.

Rewrite the integral $$\int_{0}^{1}\int_{\sqrt{x}}^{1}\int_{0}^{1-y}f(x,y,z)\,dz\,dy\,dx$$ in the orders $dx\,dy\,dz$ and $dy\,dz\,dx$. My try: We have $z=0,z=1-y,y=\sqrt{x},y=1,0\leq x\leq 1$ ...
0
votes
0answers
47 views

Even and odd integrals

Find the definite integral $$\int_{-2}^{2} \Big(2f(x) + 3g(x)\Big)dx$$ where $f(x)$ is an even function such that $$\int_{0}^{2} f(x)dx = 3$$ and $g(x)$ is such that $$\int_{-2}^{4} g(x)dx = -3 ...
1
vote
1answer
36 views

Area within $x=0$ $y=x$ and $e^{-x}$

Is there a way to find the area within $x=0$ $y=x$ and $e^{-x}$ without solving numerically $e^{-x}=x$ ?
0
votes
1answer
38 views

Integral of Euler's formula

Why is $=\int\limits_{-\infty}^{\infty}\cos(-tx)dF(x)+i\int\limits_{-\infty}^{\infty}\sin(-tx)dF(x)=\int\limits_{-\infty}^{\infty}\cos(tx)dF(x)-i\int\limits_{-\infty}^{\infty}\sin(tx)dF(x)$? I know ...
4
votes
3answers
217 views

How to evaluate $\int_{0}^{\infty}\frac{(x^2-1)\ln{x}}{1+x^4}dx$?

How to evaluate the following integral $$I=\int_{0}^{\infty}\dfrac{(x^2-1)\ln{x}}{1+x^4}dx=\dfrac{\pi^2}{4\sqrt{2}}$$ without using residue or complex analysis methods?
0
votes
1answer
20 views

Iterated integral over non-rectangular region problem.

Could we integrate $S$, where $S$ is the region bounded between $y = 1$ and $y = x^2$ over the function $xy$ by taking $y$ constant? I solve the problem by taking $x$ constant and get the result $0$. ...
3
votes
0answers
80 views

Prove that primitives of $\frac{x^3}{{\rm e}^x - 1}$ have no closed form in terms of elementary functions

It is known the following indefinite integral $$\int \frac{x^3}{{\rm e}^x - 1} dx$$ cannot be evaluated in closed form in terms of any of the elementary functions of mathematics. A proof of this can ...
1
vote
3answers
91 views

Integration of $1/(x\sqrt{25x^2-1})$

$$\int{\frac{1}{x\sqrt{25x^2-1}}}\,dx$$ Let $x=\frac{1}{5}u$ Now when I substite it and simplify I get $$\int{\frac{1}{u\sqrt{u^2-1}}}\, du$$ There is a trig identity which says that this is equal ...
7
votes
3answers
462 views

A Sum that came up while solving a integral

While evaluating $I$, I did the following- $$\begin{align}I= \int_{0}^{1} \log \left(\dfrac{1+x}{1-x}\right) \dfrac{1}{x\sqrt{1-x^2}} \ \mathrm{d}x &= 2 \int_{0}^{1}\sum_{n=0}^{\infty} ...
2
votes
2answers
74 views

Limit of an integral

I'm not sure how to approach (no pun intended) the following limit: $$\lim_{x \to 0^{+}} \sqrt{|\sin x - \tan x | } \int_{\cos x}^{1+ \sin x} e^y \, \, \mathrm{d}y$$ I know that the indefinite ...
3
votes
5answers
120 views

Trigonometric substitution and Integration of $\frac{1}{x^2\sqrt{x^2+1}} $

Regarding the integral $$ \int \frac{dx}{x^2\sqrt{x^2 + 1}} $$ I'm not sure what to do about the extra $x^2$ in the denominator. What can I do about it?
0
votes
2answers
41 views

Integrating exponential function with elliptic bounds

I am trying to integrate the following: $$\iint_R\exp\left(\frac{x^2}{4}+\frac{y^2}{16}\right)\:\mathrm{d}A$$ With the region $R$ having the bounds: $$\frac{x^2}{4}+\frac{y^2}{16}=3$$ ...
2
votes
2answers
92 views

A reduction formula for $\int_0^1 x^n/\sqrt{9 - x^2}\,\mathrm dx$

Let $$I_n = \int_0^1 \frac{x^n}{\sqrt{9 - x^2}}\,\mathrm dx$$ Using integration, show that $$nI_n = 9(n - 1)nI_{n - 2} - 2\sqrt2$$ I've found that $\displaystyle I_0 = ...
0
votes
1answer
31 views

Riemann integrable proof and notation

For Riemann integrable proof, I see $f \in \Re(\alpha)$. Also I see $U(p,f,\alpha)$. What does $\alpha$ stand for? Also to prove Riemann integrability, what do I do at very first step? I know my ...
0
votes
3answers
25 views

stuck on integrating fractions for ODE

I'm working on the ODE $\frac{dy}{dx}=\frac{xy+3x-y-3}{xy-2x+4y-8}$ I factored it $\frac{dy}{dx}=\frac{(x-1)(y+3)}{(x+4)(y-2)}$ I used separation of variables $\int \frac{y-2}{y+3} dy = \int ...
0
votes
2answers
55 views

Prove that $\frac{1}{2}ab \equiv \int_0^b \! f(x) \, \mathrm{d}x$ when calculating the area of a right triangle.

Triangle $ABC$ is a right triangle with sides $AB$, $BC$ and $AC$. $a$ is the length of $AB$. $b$ is the length of $BC$. $c$ is the length of $AC$. If $a = 3$, and $b = 4$, we can use ...
0
votes
3answers
75 views

Find $\int\sin^4(x)\cos^2(x)\,dx$

Find $$\int\sin^4(x)\cos^2(x)\,dx$$ My Attempt: $$\int\sin^4(x)\cos^2(x)\,dx = \frac18 \int ((1-\cos(2x)-\cos^2(2x)+\cos^3(2x))\, dx$$ How to proceed from here?
0
votes
0answers
18 views

Area of a smooth parametrized surface

Let $S=\mathbf{X}(D)=(x(s,t),y(s,t),z(s,t))$ be a smooth parametrized surface with $(s,t) \in\mathbf{R}^2$, where $D$ is the union of finitely many elementary regions in $\mathbf{R}^2$ and ...
0
votes
1answer
28 views

Prove the series of functions converges uniformly at $[-a,a]$ where $0<a<1$.

Let $$ \sum_{n=0}^\infty \left( \frac{x^{2n+1}}{2n+1} - \frac{x^{n+1}}{2n+2} \right) $$ Prove the series converges uniformly to $\frac{1}{2}\log(x+1)$ at $[-a, a]$ where $0<a<1$. I've ...
2
votes
0answers
42 views

Determining solid region from bounds of triple integral

If you have an integral such as: $$\int_0^1\int_0^{2-x^2}\int_0^{2-x}f(x,y,z)dydzdx$$ How can you determine the equation for the solid region represented by the bounds of this triple integral? Does ...
3
votes
2answers
78 views

How to evaluate $\int_0^1 \frac{2-t}{t^2-t+1} dt$?

How to evaluate $$\int_0^1 \frac{2-t}{t^2-t+1} dt\;?$$ I tried doing it using $s=-t+1$ but it wasn't useful. We've learned in class that having a polynomial in the denominator is considered to be ...
0
votes
0answers
83 views

Definite integral involving Error function

Let us write $$\mathrm{erf}(x)=\frac{2}{\sqrt {\pi}}\int_0^x e^{-t^2}dt $$ for the usual Gauss error function. Given natural numbers $m,n,k$ I am interested in computing the integral ...
1
vote
1answer
28 views

If $f$ is nonnegative and measurable then its integral is the limit of integrals of truncated functions

If $f$ is nonnegative and measurable, show that $$\int_{-\infty}^{+\infty} f = \lim_{n\rightarrow \infty} \int_{-n}^{n} f = \lim_{n \rightarrow \infty} \int_{\{f \geq 1/n\}} f$$ So I'm looking at ...
1
vote
2answers
100 views

Is it possible to evaluate $\int_0^1 \sin(\frac{1}{t})\,dt\,$?

I was wandering if it possible to evaluate the value of the following improper integral: $$ \int_0^1 \sin\left(\frac{1}{t}\right)\,dt $$ It is convergent since $\displaystyle\int_0^1 ...
0
votes
2answers
42 views

How to calculate the Fourier transform?

If the Fourier transform is defined by $\hat f( \xi)=\int_{-\infty}^{\infty}e^{-ix \xi}f(x)dx$. How to calculate the Fourier transform of $$\begin{equation*} f(x)= \begin{cases} ...
1
vote
4answers
69 views

Integrals are equal

Suppose that $f$ is integrable on $[a, b]$. Prove that there is a number $x \in [a, b]$ such that $$\int_a^x f(t)\,dt = \int_x^b f(t)\,dt .$$ Show by example that it is not always possible to choose ...
0
votes
1answer
14 views

$\int_C (\alpha x, -\alpha y) . dr = 0$ where C is the unit circle

Circulation is given by $$\int_C u . dr$$ I want to show that the circulation around the unit circle is $0$ for $u = (\alpha x, \alpha y)$. Ie. $$\int_C (\alpha x, -\alpha y) . dr = 0$$ How would ...