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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1answer
98 views

Compute $\int \frac{\sin x+\cos x}{\sqrt[3]{\sin x-\cos x}}\,\mathrm dx$

Compute $$\int \frac{\sin x+\cos x}{\sqrt[3]{\sin x-\cos x}}\,\mathrm dx$$ Let $t=\tan\frac x2$, then $$\sin x=\frac{2t}{1+t^2},\; \cos x=\frac{1-t^2}{1+t^2},\; \mathrm dx =\frac{2\mathrm ...
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3answers
65 views

Area of the circle.

I want to calculate the area of the circle of radius $\mathfrak{R}$. I would like to do it using the Cartesian coordinates (not the polar ones). The problem is that I found the area of a circle of ...
0
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1answer
88 views

Summation of integrals over odd and even parts

Let the function $f: [0,1]\rightarrow \mathbb{R}$ be continuous. For $n\in \mathbb{P}$, partition $[0,1]$ into $2n$ equal pieces, and add up the integrals over the odd and even parts separately. ...
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0answers
52 views

Generalization of Gauss' mean value theorem

EDIT: How to get this, if it is possible to get it: $$f^{(n)}(z_o)=\frac{n!}{2\pi}\int_0^{2\pi}f(e^{i\theta}+z_o)(e^{ni\theta})^{-1}\ d\theta$$ I was reading this question, and wanted to write it ...
2
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1answer
63 views

Coordinates of tilted circle.

The original question is as follows: Imagine a wire located at the intersection of $x^2+y^2+z^2=1$ and $x+y+z=0$, whose density depends on position according to $\rho({\bf x})=x^2$ per unit length. ...
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1answer
91 views

About the confluent versions of Appell Hypergeometric Function and Lauricella Functions

I know there are two important properties about Appell Hypergeometric Function and Lauricella Functions: ...
2
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1answer
52 views

Area of a part of a cylinder contained in a ball

How can one calculate area of the part of a cylinder $x^2+y^2=r^2$ which is contained inside this ball : $(x-r)^2 + y^2 + z^2 \le 4r^2$? I've read a little about surface integrals, but I don't know ...
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2answers
31 views

Area of $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$

Could you tell me how to calculate the area of part of the plane: $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$, $a, b, c >0$ where all coordinates of a point are positive?
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2answers
93 views

Definite Integral of $e^{\large x^2}$

I know there's no elementary antiderivative of $e^{\large x^2}$. But what if there's a definite integral like $$\int_0^1e^{\large x^2}\ dx\ ?$$ I tried using basic definite integral property like ...
2
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3answers
110 views

How can ${\iint\limits_{D}{{e^{x^2+y^2}}}}dxdy$ be found?

How can ${\iint\limits_{D}{{e^{x^2+y^2}}}}dxdy $ be found, if $D$ is $x$ O $y$ axis? So far I have done it this far: ...
2
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2answers
68 views

Limit of a function with a defined integral

I have the next limit: \begin{equation} \lim_{x\to\ 0}\displaystyle\frac{\displaystyle\int_0^{x^2}{\frac{1-\cos{t^2}+at^4}{t}}dt}{(1-\cos{(\frac{x}{3})})^4} \end{equation} I've tried to solve it by ...
3
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1answer
72 views

Solving defined integral

Is there an analytical solution to the following integral: $$ I = \iint\limits_{\mathcal{D}} \exp\left(-kx\right) \mathrm{d}x \mathrm{d}y $$ Where: $$ \mathcal{D}(x,y) \equiv x^2 + y^2 \leq R^2 $$ ...
1
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1answer
52 views

integrating $x^e$

I know the rule in which the base to power $x$ will always remain the same when integrating or differentiating however I am unsure how to deal with $x^e$. Does the rule also apply to this; otherwise ...
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2answers
63 views

Integration over the cube

I have the following exercise: Integrate the $g=x \cdot y \cdot z$ over the cube that is on the first octant and that is bounded from the levels $x=1, y=1, z=1$. Could you give me some hint what I ...
1
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1answer
42 views

Integral convergence with parameter

Study for which $\beta>0$ the following integral converge: $$\int_0^\infty \frac{\arctan(\ln^2x)}{\left|\beta-x\right|^e} dx$$ My try: I managed only to see that: $$\int_a^\infty ...
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2answers
103 views

How can I find the integral?

I want to find the integral $$\int_{0}^{1}{\sqrt[3]{2x^3-3x^2-x+1}}\,\mathrm{d}x.$$ I tried, $$\int_{0}^{1}{\sqrt[3]{(-1 + 2 x) (-1 - x + x^2)}}\,\mathrm{d}x.$$ Put $t =-1 - x + x^2$, then $\mathrm{d} ...
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2answers
21 views

How can I linearize annual sales forecasts into a sequential number of series, while maintaining contraints on area under the curve

Run into a problem at work with forecasting and thought people here might be able to help. Our sales team have provided me product forecasts in the form of annual volumes (eg. FY14=100M units, ...
4
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3answers
124 views

How to solve $\int \frac{x^4 + 1 }{x^6 + 1}$?

How to solve $\int \frac{x^4 + 1 }{x^6 + 1}$ ? The numerator is a irreducible polynomial so I can't use partial fractions. I tried the substitutions $t = x^2, t=x^4$ and for the formula $\int ...
0
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0answers
43 views

Closed form of parametric integral

I'm looking for a closed form for the parametric integral $f(a) = \int_0^1 (ax + b)^k dx$. If $a\neq 0$, $f(a)$ is a rational function $$f(a) = \frac{(a+b)^{k+1} - b^{k+1}}{a(k+1)}$$ which reduces to ...
4
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1answer
334 views

Gaussian Matrix Integral

I need your help to solve this exercise : Let $S$ be a symmetric Hermitian matrix $N\times N$ : $S=(s_{ij})$ with $s_{ij}=s_{ji}$. When $\langle s_{ij}s_{kl}\rangle\neq 0$ What $$\int ...
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1answer
53 views

Questions about the line integral

Here's how we get to the formula for the line integral: $$\overrightarrow{R}(t)=x(t) \hat{\imath}+y(t) \hat{\jmath}+z(t) \hat{k}, \ \ \ \ \ \ a \leq t \leq b$$ We subdivide the curve into the ...
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1answer
53 views

Computing the derivative of an integral

There are similar questions on the same topic, yet I could not figure out why the following equation (taken from an economics solution manual) holds: $$ \frac{\partial}{\partial C(i,j)} ...
3
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1answer
56 views

Asymptotic approximation to find the Barnes integral

In the following paper and in order to prove the Barnes integral $$\frac{1}{2\pi ...
3
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1answer
133 views

How to reproduce the Mathematica solution for $\int(\cos x)^{\frac23}dx$?

I entered this integration problem to Mathematica Online Integrator an got a solution I would never have been able to find manually. $$\int\root 3 \of{\cos(x)^2}\,dx=\frac{(-3\cos(x)\root 3 ...
2
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1answer
62 views

Questions about the surface integral

Here's how we get to the formula for the surface integral: $$\Delta P_k=\frac{\Delta A_k}{\cos{\gamma}}$$ $$g:\text{ density }$$ $$\text{ Integral }=\sum_k \Delta P_k \cdot g(x_k, y_k, z_k) ...
3
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3answers
154 views

How to integrate $\int_0^\pi \frac{1}{\sqrt{1+k^2\sin^2\phi}} d \phi$?

I am currently dealing with the integral $$\int_{0}^{\large\pi}\frac{{\rm d}\phi} {\,\sqrt{\vphantom{\Large A}\,1 + k^{2}\sin^{2} \phi \,}\,} $$ I know that if I had a minus sign in the denominator, ...
1
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2answers
77 views

Procedure for evaluating $\int_{x=\ -1}^1\int_{y=\ -\sqrt{1-x^2}}^{\sqrt{1-x^2}}\frac{x^2+y^2}{\sqrt{{1-x^2-y^2}}}\,dy\,dx$

While solving another problem I have come across this integral which I am unable to evaluate. Can someone please evaluate the following integral? Thank you. $$\int_{x=\ -1}^1\int_{\large y=\ ...
0
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1answer
32 views

derivate of indicator function

What is the derivative of the indicator function: \begin{equation} f(x)=\begin{cases} 1 & x^{\min} x\leq x^{\max}\\ -\infty &\mbox{otherwise}? \end{cases} \end{equation} thank you
2
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1answer
65 views

limit of the integrations of a sequence of integrable functions

Let $(f_n)^\infty_{n=1}$ be a sequence of Lebesgue integrable functions on $[0,1]$ such that $f_n$ converges to $f$ almost everywhere in $[0,1]$. Suppose further (a). ...
9
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4answers
257 views

integral with tanh and sin

Here may be a fun one to try. show that: $$\int_{0}^{\infty}\frac{\tanh(\frac{\pi x}{2})\sin(ax)}{x^{2}+1}dx=1/2\left[e^{-a}\log(e^{2a}-1)-e^{a}\log(1-e^{-2a})\right]$$
1
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2answers
62 views

How do I integrate $\frac{1}{x}$ when $x$ has a power?

As the title says, how would I integrate $\frac {1}{x^2}$? I know that $\frac {1}{x}$ integrates to $\ln x$ but I am unsure when $x$ has a power.
1
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5answers
94 views

Complex integral - exercise

$$ \int\limits_{C(-2, \frac{1}{4})} = \frac{e^z}{z^2-4}dz$$ C is a circle center = -2 and radius = $\frac{1}{4}$ z is a complex number I don't know how to do the exercises like that.
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0answers
32 views

Statistical Integration by part

Would it be possible to give me some hint( such as integration by part or any other method ) that how can I prove that left hand side is equal to the right hand side? Thank you in advance ...
1
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1answer
31 views

questions about $L^p$ space with $0<p\leq 1$ parallel to the case $1<p$

Question (1). Riesz-Fischer Theorem: For $1\leq p\leq \infty$, $L^p(\mu)$ is complete. Corollary of proof: Let $1\leq p\leq \infty$. If $(f_n)_{n=1}^\infty$ is a sequence coverging to $f$ with ...
1
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2answers
53 views

Partial Fractions Integration Question

$$\int\frac{x^5+x-1}{x^3 +1} dx$$ Have tried everything ... polynomial long division, partial fractions, trig substitution etc... Not for an assignment, so if a complete solution could be provided ...
6
votes
3answers
190 views

Integral $\iint \limits_{{x,y \ \in \ [0,1]}} \frac{\log(1-x)\log(1-y)}{1-xy}dx\,dy=\frac{17\pi^4}{360}$

Hi I am trying to integrate $$ \mathcal{I}:=\iint \limits_{{x,y \ \in \ [0,1]}} \frac{\log(1-x)\log(1-y)}{1-xy}dx\,dy=\int_0^1\int_0^1 \frac{\log(1-x)\log(1-y)}{1-xy}dx \,dy $$ A closed form does ...
5
votes
1answer
108 views

How to calculate the integration $\int_{0}^{\pi}\frac{dx}{(2-\cos{x})^2}$ [duplicate]

Given that $$ \int_{0}^{\pi}\frac{dx}{2-\cos{x}}=\frac{\pi}{\sqrt{a^2-1}} $$ How to calculate the integral $$ \int_{0}^{\pi}\frac{dx}{(2-\cos{x})^2} $$
0
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1answer
55 views

Using Poisson's integral formula

The problem asks to prove the following equality using Poisson's integral formula (or Poisson kernel, if I understood correctly from Wikipedia): $$\int_0^{2\pi} \frac{e^{\cos ...
2
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1answer
43 views

Scale-invariance of $\int_0^\infty \frac{f(x)}{x} \ dx$

Let $f$ be some non-negative, measurable function on $[0,\infty)$. The quantity $\int_0^\infty \frac{f(x)}{x} \ dx$ is scale-invariant in the sense that, if one puts $f_c(x) := f(cx)$ for $c > 0$, ...
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0answers
45 views

Integration notation

could someone explain the following notation: $u(x,t):=\int^t_0 v(x,t:\tau)d \tau$ It's come up but I don't understand how to interpret the semicoloned tau
1
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1answer
139 views

Double Integral $\iint_D\ (x+2y)\ dxdy$

$$\iint_D (x+2y)\ dxdy $$ If the area is range by $x=2,\ x=3,\ y=x,\ y=2x$, how to include the lines? How limits for integral will looks like? You mean something like this? ( I made mess) $$\iint_D ...
2
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1answer
51 views

how to calculate this double integral

I have this double integral: $$ \int_0^\frac{\pi}{4}\int_0^\frac{\pi}{2}(\cos x + \sin y) \, dy \, dx $$ Is this correct? $$\begin{align*} \int_0^\frac{\pi}{2}(\cos x + \sin y) \, dy &= ...
2
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1answer
67 views

Change of variable

I have to approximate the following integral, using Simpson's Composite $1/3$ Rule: $\displaystyle \int\limits_{0}^1 \mathrm{\frac{e^{2x}}{\sqrt[5]{x^2}}}\,\mathrm{d}x$. The only problem is that ...
1
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1answer
74 views

A problem involving Stieltjes Integral and bounded variation

I found this problem in a book I'm using to study (Curso de Análise - Vol 2, Elon Lages Lima). "Let $\alpha:[a,b] \to \mathbb{R}$ be a bounded function. If $\displaystyle \int_{a}^{b}f(t)d\alpha$ ...
1
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1answer
41 views

How to take the derivative of a function $F(x)$

The function $F(x)=\int_{-1}^{x}\sqrt{1-t^2}dt$. I believe this to be the representation of the area under the curve between $-1$ and $x$, where $\int_{-1}^{x}\sqrt{1-t^2}dt$ is a function of $x$: ...
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2answers
40 views

Complex integration around a singularity [duplicate]

I am trying to integrate the function $f(z)=$$\frac{5}{z^2}$ from -3 to 3 and I am supposed to develop a closed region that avoids the origin and use the analyticity of the function in this region to ...
4
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4answers
361 views

Prove that $\int_a^c f(t)dt - \int_c^b f(t)dt = f(c)(a+b-2c) $, for some $c\in(a,b)$

Let $f$ be a continuous on $[a,b]$ then prove that there exist some $c$ that lies in $(a,b)$ such that $$\int_a^cf(t)\,dt - \int_c^b f(t)\,dt = f(c)(a+b-2c) $$ and hence prove that $\int_a^c ...
2
votes
4answers
127 views

Understanding this calculus simplification

I'm having a lot of trouble computing answers in the "Arc Length and Surfaces of Revolution", and I found it probably has to do with me not understanding the following kinds of simplication: $$S= ...
11
votes
6answers
365 views

Evaluating the integral with trigonometric integrand

While solving another problem I have come across this integral which I am unable to evaluate. Can someone please evaluate the following integral? Thank you. ...
0
votes
0answers
33 views

Show if this is integrable (defined 1 on rationals, 0 else)

Define $f: [0,1] \rightarrow \mathbb{R}$ as $f(x) = \begin{cases} 0 & x \in \mathbb{Q} \\ 1 & x \notin \mathbb{Q} \end{cases}$ Find $\underline{\int_0^1f}$ and $\overline{\int_0^1f}$. Is ...