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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1
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13 views

A vertical dam has a semicircular gate

A vertical dam has a semicircular gate. Find the hydrostatic force against the gate. The dam is 12 meters high, the water level is at 10 meters, and the semicircular gate had a diameter of 4 meters. ...
0
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1answer
53 views

evaluating $ \int\limits _{0}^{1}\frac{1}{\sqrt{x+\varepsilon}}dx $

I came across this : I'm trying to evaluate it up to $ o(\epsilon) $ $$ F\left(\varepsilon\right)=\int\limits _{0}^{1}\frac{1}{\sqrt{x+\varepsilon}} \, \mathrm{d}x $$ I've trying considering to look ...
1
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1answer
66 views

Show that $\lim_{x\to\infty} f(x) = 0$.

Let $f\in C^1$. Let's assume that $\int_0^\infty f(x)\ dx$ converges and $f'(x)$ is bounded. Prove that $\lim_{x\to\infty} f(x) = 0$. Let's assume by contradiction that $\lim_{x\to\infty} f(x) ...
2
votes
1answer
125 views

Integration by substitution - where is the mistake?

I want to integrate $$\int_{-1}^{1} (1-x^2)^{3/2} \, \mathrm{d}x$$ by substituting $x=\cos z$ and $dx = -\sin z \, dz$. $x=-1 \implies z=-\pi $ and $x=1 \implies z=0$. I receive: ...
1
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4answers
84 views

Is integration of $x\operatorname{cosec}(x)$ defined?

Is integration of $x\operatorname{cosec}(x)$ possible? If yes, then what is its closed form; if not, then why is it non-integrable ?
2
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2answers
39 views

An improper integral and its convegence

I have an integral $$I(\gamma)=\int\int d^3 \mathbf{r} \, d^3 \mathbf{r}' \frac{1}{|\mathbf{r}-\mathbf{r}'|+\gamma}$$ were $\gamma$ is a positive number, $\mathbf{r},\mathbf{r}' \in \mathbb{R}^3$, ...
0
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1answer
23 views

Is the integral with respect to increasing continuous functions the limit of integrals with respect to $C^1$ functions?

if $\xi$ is continuous increasing can we find $\xi^n\in C^1$ such that $$\int_0^t f(u)\, d\xi = \lim_n\int_0^t f(u)\, d\xi^n$$ for every continuous $f$?
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3answers
64 views

Integral of rational function with a squared term in the denominator

I know the integration when in the reciprocal there's only degree $1$, but what about degree $2$? Take an example, $$\int\frac{x \, \mathrm{d}x}{a+bx^2}$$
1
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1answer
20 views

Bounds for double exponential integrals

I understand that the double-exponential integral $$ F(a,b,C) := \int_{C}^\infty \exp(-a \exp(b x)) \, dx \quad \text{(with $a,b>0$ and $C \geq 0$)} $$ can in general not be solved in closed-form. ...
0
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2answers
43 views

Evaluating the integral in polar

I am trying to show that the double integral of $\sqrt{\rho^2-y^2}$ for $x$ between $0$ and $\rho$ while $y$ is between $0$ and $\sqrt{\rho^2-x^2}$ is $(2/3)(\rho)$. In cartesian I have tried its ...
0
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1answer
28 views

Area of region - double integral

Here is my task: Calculate area of region $(x^{2}+y^{2})^{2}\leq a^{2}(x^{2}-y^{2})$. Here is what I have done. After transforming this line to polar form $(x=\rho\cos\phi,y=\rho\sin\phi)$, we have: ...
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2answers
32 views

Let $f(x) = [x], x \in [1,3]; \ \phi(x) = x , x \in [1,2]$ and $= 2x -2, x \in (2,3]$.show that $\int_1^3 f = \phi(3) - \phi (1)$

Let $f(x) = [x], x \in [1,3]; \ \phi(x) = x , x \in [1,2]$ and $= 2x -2, x \in (2,3]$. Then to show that $f$ is integrable and evaluating the value of $\int_1^3 f$. I have done upto this. But ...
0
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2answers
44 views

Fundamental Theorem of Calculus with 1/lnx

I'm struggling with this problem, because I'm not sure how to integrate $1/\ln(x)$ Suppose that you have the following information about a function $F(x)$: $$F(0)=1, F(1)=2, F(2)=5$$ ...
1
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0answers
40 views

Fractional part of $n\alpha$ is equidistributed

Let $\alpha$ be an irrational number. Then the sequence $\{\{n\alpha\}\}$ is equidistributed. I am using the following definition of equidistribution. A sequence $\{a_i\}$ is equidistributed if ...
1
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2answers
50 views

Integration of two different functions

I'm stuck with a problem, can anyone help? What if we integrate $$\int \frac{x\cdot dx}{mg+kx}$$? Actually I faced this in physics. But I need this basic knowledge of integration. Can anyone help?
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0answers
39 views

Need help for calculating two derivatives analytically

During building a Jacobian matrix for a numerical simulation I need to calculate following two derivatives where I hesitate about the correct answer: $$\partial( \partial X/ \partial\theta)/\partial ...
1
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1answer
40 views

Integral by parts?

Let $f:{\mathbb R}\rightarrow {\mathbb R}_+$ be a density function with finite expectation. This is, $$\int_{\mathbb R}x f(x)dx<\infty.$$ Suppose that we want to integrate $I(a)=\int_a^{\infty} x ...
0
votes
2answers
34 views

What to do in this Surface Integral?

Calculate the surface integral: $$\iint_\sigma f(x,y,z)\ \mathrm{d}S$$ Where: $f(x,y,z) = x-y-z$ and $\sigma$ is the portion of the plane $x+y=1$ on the first octant between $z=0$ e ...
1
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0answers
34 views

An intriguing integral $\int \ln{|\nabla u|^2}\, \mathrm{d}u$

How to integrate $\int \ln{|\nabla u|^2}\, \mathrm{d}u$ ? In Cartesian co-ordinates this would be $\int \ln{(u_x^2 + u_y^2)}\, \mathrm{d}u$, where $u_x \equiv \frac{\partial u}{\partial x}$. We ...
1
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1answer
48 views

let $\phi (x) =\lim_{n \to \infty} \frac{x^n +2}{x^n +1}$; and $f(x) = \int_0^x \phi(t)dt$. Then $f$ is not differentiable at $1$.

For $x \geq 0$, let $\phi (x) = \lim_{n \to \infty} \frac{x^n +2}{x^n +1}$; and $f(x) = \int_0^x \phi(t)dt$. Then $f$ is continuous at $1$ but not differentiable at $1$. First we calculate $\phi (x) ...
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1answer
28 views

How to evaluate this combination of sums and integrals?

I am reading a book on PDEs, and I am near the beginning where the author is talking about the heat equation and, specifically, solving the non-homogenous equation $u_t={\alpha}^2u_{xx}+f(x,t).$ The ...
1
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1answer
22 views

Prove that $\left\{u\in W_0^{1,2}(\Omega):\int_\Omega|u|^{p+1}\;d\lambda^n=1\right\}$ is well-defined and closed

Let $\Omega\subseteq\mathbb{R}^n$ be a domain with a smooth boundary $H:=W_0^{1,2}(\Omega)$ be the Sobolev space $p>1$ such that $$p<\begin{cases}\infty&\text{, if ...
3
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2answers
93 views

Computing $\int_{0}^{1\over 2}{\ln(1+x)\over x}dx$.

Compute $\int_{0}^{1\over 2}{\ln(1+x)\over x}dx$ with a precision (Accuracy? Error? What is the formal expression?) of 0.01. Attempt: First of all: $\ln(x+1)=\sum_{k=1}^{\infty}{(-1)^{k-1}x^k\over ...
6
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1answer
46 views

Finding a better upper bound for an integral of a product of $n$ terms

So I'm trying to find and upper bound for the integral $$ \int\limits_{a}^b \! (x-x_1)^2 \cdots (x-x_n)^2\, \mathrm{d}x, $$ where $x_i \in [a,b], \enspace \forall i=1,\dots ,n.$ I've tried ...
0
votes
0answers
34 views

If $u \in L^2(0,T;L^2(\Omega))$ is $\int_{\Omega}\int_0^T |u(t,x)|^2$ defined?

Let $u \in L^2(0,T;L^2(\Omega))$ on some domain $\Omega$. We know that $$\int_0^T \int_{\Omega}|u(t,x)|^2$$ is defined, but is it equal to $$\int_{\Omega}\int_0^T |u(t,x)|^2?$$ Can I interchange the ...
2
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1answer
85 views

Investigate the convergence of $\int _0^\infty \frac{\sin x^2}{x} \ dx$

Investigate the convergence of $$\int_0^\infty \frac{\sin x^2}{x} \, \mathrm{d}x$$ Is it converging? Converging absolutely? I want to use Dirichlet's test for integrals. Let $f(x) = \frac 1 x$ ...
2
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0answers
36 views

“Triangle” inequality for integrals

I have got two questions: 1) Let $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ be any continuous function. Let $\Gamma$ be a piecewise smooth curve on $\mathbb{R}^2$. The following inequality holds: ...
1
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0answers
36 views

Why is $\sum_{n=0}^{N}{\cos nx}={1\over 2}+{1\over 2}\sum_{-N}^{N}e^{inx}$?

Why is $\sum_{n=0}^{N}{\cos nx}={1\over 2}+{1\over 2}\sum_{-N}^{N}e^{inx}$? I have gone through all the identities relating Fourier series and I can't seem to understand why. In this question, the ...
0
votes
2answers
35 views

How can I solve this integral with the comparison theorem?

I have an integral that I am not sure how to solve with the comparison theorem to see if it is divergent or convergent. $$\int_1^\infty\frac{e^{-2x}}{\sqrt{x+16}}\;dx$$ How can I solve this with ...
2
votes
2answers
34 views

How to solve the integral $\int\tan^{3}x \sec^{3/2}x\; dx$?

How to solve the following indefinite integral $$\int \tan^{3}x \sec^{3/2}x \; dx$$ to get the solution in the form of $$\large\frac{2}{7}\sec^{7/2}x - \frac{2}{3}\sec^{3/2}x +c$$ I tried taking ...
2
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1answer
66 views

How to prove that $\frac 12+ \frac 13+\dots + \frac 1n < \log n < 1 + \frac 12+ \dots + \frac {1}{n-1} $?

If $n \in \mathbb N$ and $n \geq 2$, then we have $\frac 12+ \frac 13+\dots + \frac 1n < \log n < 1 + \frac 12+ \dots + \frac {1}{n-1} $. My try : Once if we can prove that for all $k \in ...
1
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1answer
33 views

Evaluating $\int_{\gamma} \frac{z}{\cosh (z) -1}dz$

Evaluate $\int_{\gamma} \frac{z}{\cosh (z) -1}dz$ where $\gamma$ is the positively oriented boundary of $\{x+iy \in \Bbb{C} : y^2 < (4\pi^2 -1)(1-x^2)\}$. I just learned the residue theorem, ...
0
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2answers
43 views

$f: [0,1] \to \mathbb R$ is continuous and $\int_0^x f(t) dt = \int_x^1 f(t)dt$ for all $x \in [0,1]$, then $f(x) = 0$ for all $x \in [0,1]$.

A function $f: [0,1] \to \mathbb R$ is continuous on $[0,1]$ and $\int_0^x f(t) dt = \int_x^1 f(t)dt$ for all $x \in [0,1]$, then $f(x) = 0$ for all $x \in [0,1]$. My Try: Let us assume that $f(x) ...
0
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2answers
65 views

How to integrate $\int\frac{3x+2}{x^2-x-2}dx$

This is the indefinite integral I have to evaluate: $$\int\frac{x^3}{x^2-x-2}dx$$ so by using the long division on polynomials technique, I got to: $$\frac{x^2}{2}+x+\int\frac{3x+2}{x^2-x-2}dx$$ How ...
7
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4answers
172 views

Calculate $\int _0^\infty \frac{\ln x}{(x^2+1)^2}dx$

Calculate $$\int _0^\infty \dfrac{\ln x}{(x^2+1)^2}dx.$$ I am having trouble using Jordan's lemma for this kind of integral. Moreover, can I multiply it by half and evaluate $\frac{1}{2}\int_0^\infty ...
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0answers
32 views

Solutions of the following differential equation [on hold]

$$\frac{-2q}{k}+z^2+2zp-2zN+(p-N)^2=0$$ What is the solution of this differential equation? Where $N$ is a constant and $p$ and $q$ are the usual notations.
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0answers
50 views

Can any one help me solve this integral ??? [on hold]

![i cannot able to solve this integral ,can any one able to solve this integral and i used integral technique but i cannot able to solve this equation the integral is with respect to x ...
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1answer
104 views

Evaluate the following Integration--

Evaluate the following Integration $$\int \frac{\cos^9 x}{\sin^3 x + \cos^3 x} \,dx$$ I tried, but this problem is very difficult to me. any help?
4
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1answer
62 views

$f(x) =\lim_{n \to \infty} \frac{(1+ \sin \frac{\pi}x)^n - 1} { (1+ \sin \frac{\pi}x)^n +1}$, $x \in (0,1]$. To show that $f$ is integrable on $[0,1]$

A function defined on $[0,1]$ by $f(0) = 0$ and $f(x) = \lim_{n \to \infty} \frac{(1+ \sin \frac{\pi}x)^n - 1} { (1+ \sin \frac{\pi}x)^n +1}$, $x \in (0,1]$. To show that $f$ is integrable on ...
2
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1answer
52 views

Solving this Integral with Bessel Functions

Any suggestions on solving this (J0,J1 Bessel function of first kind, 0th and 1st order, respectively) : $$ T = \int_0^a \int_0^\infty J_0(\lambda r) J_1(\lambda a) ...
0
votes
0answers
19 views

Area of region - Double integrals

Here is my task: Calculate area of region $(\frac{x}{a}+\frac{y}{b})^{5}=\frac{x^{2}y^{2}}{c^{4}}$,$a,b,c>0$. Solution is $A=\frac{a^{5}b^{5}}{1260c^{8}}$ Any idea how to solve this?
4
votes
3answers
89 views

Why is $\int_{0}^{\pi}{1\over 1-\sin x}dx=2\int_{0}^{\pi\over 2}{1\over 1-\sin x}dx$?

Why is $\int_{0}^{\pi}{1\over 1-\sin x}dx=2\int_{0}^{\pi\over 2}{1\over 1-\sin x}dx$, or to be accurate: why is $\int_{\pi\over 2}^{\pi}{1\over 1-\sin x}dx=\int_{0}^{\pi\over 2}{1\over 1-\sin x}dx$? ...
1
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0answers
15 views

Prove that $\frac{\langle f^2,g\rangle_{L^2}}{\left\|f\right\|_{L^2}^2}\ge-\left\|g\right\|_{L^\infty}$ for $f\in L^2$ and $g\in L^\infty$

Let $\Omega\subseteq\mathbb{R}^n$ be bounded, $f\in L^2(\Omega)$ and $g\in L^\infty(\Omega)$. How can we show, that $$\frac{\langle ...
0
votes
2answers
32 views

(definite integral) area between two trig functions

I'm trying to figure out how to find the area between two trig functions. I know the procedure of integration here, finding the difference between two functions and integrating across whatever ...
0
votes
1answer
26 views

How would you integrate this homogeneous equation?

I am solving a homogeneous equation $\frac{dy}{dx}= \frac{x^2+xy+y^2}{x^2}$ and have come to this step and I'm stuck now with the integration. I could really use some helpful hints to help me $$ ...
2
votes
2answers
127 views

Evaluate the improper integral $\int_{0}^{\infty}{f(x)-f(2x)\over x}dx$, where $\lim_{x \to \infty} f(x) = L$ [duplicate]

Find $$\int_{0}^{\infty}{f(x)-f(2x)\over x}\, \mathrm{d}x$$ if $f\in C([0,\infty])$ and $\lim\limits_{x\to \infty}{f(x)=L}$. I tried denoting $\displaystyle \int{f(x)\over x}dx=F(x)$, but I don't ...
6
votes
4answers
208 views

Help with the contour for this integral using residues

$$ PV \int_0^\infty \frac{dx}{\sqrt{x}(x^2-1)} $$ A keyhole contour can't be used because we have a pole in the real positive axis, isn't it?
0
votes
0answers
26 views

Integral gaussian hypergeometric function

How can we define integral with interval $[b,\infty)$ $$ \begin{align} C(b,\alpha) & = \int_b^\infty \frac{1}{1+w^{\alpha/2}}\,\mathrm{d}w \\[8pt] & = 2\pi/\alpha \csc(2\pi/\alpha)-b_2 F_1 ...
5
votes
1answer
118 views

Definite integral with logarithm and arctangent inside of arctangent

How to prove $$\int_0^1 \left[ \frac{2}{\pi }\arctan \left(\frac 2 \pi \arctan \frac{1}{x} + \frac{1}{\pi }\ln \frac{1 + x}{1 - x}\right) - \frac{1}{2} \right]\frac{\mathrm{d}x} x = \frac{1}{2} \ln ...
0
votes
1answer
49 views

Integrate the function by substitution method.

$$\int \frac1{ \cos(x-a)\cos(x-b)} \, \mathrm{d}x$$ Can someone help me to integrate this function by method of substitution.I am not able to start it for possibilities are not coming in my mind. ...