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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Evaluating and Simplifying a Double Integral

I have an integral as follows $f(t) = \int_r^\infty \frac{(sP)^{1-\rho}t^{-\alpha/2}}{1+(sP)^{1-\rho}t^{-\alpha/2}} \;dt$ I wish to get rid of the $s$ in $f(t)$ because this is an inner integral ...
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0answers
9 views

Normal convolved to Exp(polynomial)?

Is there an analytic solution for a Normal (normalized Gaussian) distribution of variance v convolved to e^y(x), where y(x) is an m-th order polynomial? Assume that m is even and the m-th coeff of y ...
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0answers
38 views

Help in solving an integral.

I am trying to evaluate this integral, but could not find a solution. I tried it, assuming it to be product of two exponential and then tried integration by parts but it does not lead to anywhere. Can ...
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3answers
75 views

Proving $\sin^2(x) + \cos^2(x) =1$ using calculus

Ok so the book in which I found this doesn't say mention the trigonometric functions by name but the question is: Let $s(x)$ and $c(x)$ be functions satisfying $s'(x)=c(x)$ and $c'(x)= -s(x)$ for ...
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2answers
26 views

How to solve an integral with the use of arcsine

The specific question is the following, $$\int_{-a}^x \sqrt{a^2-x^2}\,dx$$ We are also given that $0\le x\le a$ Thank you very much for helping.
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0answers
32 views

is it possible to evaluate any definite integral using the definition of the definite integral?

I was evaluating definite integrals using the fundamental theorem, however, out of curiosity, I wanted to see if it was possible to evaluate the following, using the definition of the definite ...
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12 views

upper-band of the Integral expression

Consider below integral expression $$\int_{0}^{\infty}g(y)[\int_{a}^{\infty}(1-e^{-(k+y)x})f(x)dx ]dy \ \ \ \ (1)$$ Where, we know: $$f(x)>0\ ,\ \ a\leq x \leq \infty$$ $$\ k>0$$ $$g(y)>0\ ...
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3answers
27 views

Help with a derivative of integral please.

I'm supposed to calculate the derivative of $\frac{d}{dx}\int_{x^{2}}^{x^{8}}\sqrt{8t}dt$ the answer I got is $8x^7\cdot \sqrt{8x^8}$ but when I put this into the grading computer it is marked wrong. ...
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0answers
11 views

Calculate the flux through a closed surface

While studying for a test I have encountered such a task: Calculate the flux through a closed surface, where $S$ is a boundary of area $V$ with an outward orientation. The data: ...
5
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1answer
68 views

A triple integral dancing in the unit cube

Straight integration seems pretty tedious and difficult, and I suppose that the symmetry might possibly open some new ways of which I'm not aware. What would your idea be? $$\int_0^1 \int_0^1 ...
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0answers
31 views

Different representations of Appell hypergeometric series

The (first) Appell series: $$F(a; b_1, b_2; c \mid z_1, z_2) = \sum_{n=0}^{\infty} \frac{(a)_n}{(c)_n} \sum_{n_1+n_2=n} (b_1)_{n_1} (b_2)_{n_2} \, \frac{z^{n_1}}{n_1!} \frac{z^{n_2}}{n_2!}$$ can be ...
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0answers
60 views

Evaluate $\int \dfrac{1}{\sqrt{x-1}+\sqrt{x}+\sqrt{x+1}} \ \mathrm{d}x$ [duplicate]

Evaluate $$\int \dfrac{1}{\sqrt{x-1}+\sqrt{x}+\sqrt{x+1}} \ \mathrm{d}x$$ I tried rationalizing the denominator by twice multiplying, but it didn't do any good. I also tried trig ...
2
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2answers
76 views

Integral: $\int \sqrt {\sin x} \, \mathrm{d}x$

I want to find $$\int \sqrt {\sin x} \, \mathrm{d}x$$ Now what I think that this can not be integrated without any definite boundary given o.w we can shhift it to gamma function or directly using ...
2
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1answer
77 views

Evaluation of $ \int_{-\infty}^{\infty}\arctan (\frac 1{2x^2})\ \mathrm dx$

Evaluate $$\int_{-\infty}^{\infty}\arctan\left(\frac{1}{2x^2}\right)\mathrm dx$$ And how can I solve it using $$\sum^{\infty}_{x=-\infty}\arctan\left(\frac{1}{2x^2}\right)\quad\text{ and ...
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1answer
16 views

Derivative of Poisson that approximates Binomial

Instead of a standard urn ball problem, I have many urns and balls. Many. One might say, a continuum of balls $B$ and urns $U$. The likelihood of a single urn having $x$ matches is, under the ...
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0answers
14 views

Trapezoidal Rule yielding the exact value of the integral

It is clear that if a function $f(x)$ is linear over the domain $a \leq x \leq b$, then one application of the trapezoidal rule, over the same domain, will yield the exact value of ...
2
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4answers
47 views

Integral $\int{ \frac{1}{\sqrt {1 - e^{2x}} } dx}$

I need a hint how to start solving this integral: $$\int{ \frac{1}{\sqrt {1 - e^{2x}} } dx}$$
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2answers
49 views

Do nested integrals exist?

I have a problem that involves evaluating (or at lest simplifying) the expression $$\int_{0}^{x}\int_0^{x'}f(y)dy dx'.$$ Playing around with Riemann sums has lead me to believe that this is just ...
1
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1answer
30 views

Approximaing Gamma function

For $c>1$ and $0<\theta<1$, we wish to approximate (upper bound) following Gamma function: $$\int_c^{c\theta}x^{-3}e^{-x}dx $$
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4 views

Lebesgue-integrability of piecewise function with random variable

This function is Lebesgue-integrable:$$\chi(x)= \left\{ \begin{array}{ll} 1 & \text{if}~x~\text{is rational}\\ 0 & \text{if}~x~\text{is irrational}. \end{array} ...
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2answers
41 views

$f \in C[a,b]$ be such that $\int_c^d f(x)dx=0 , \forall c,d \in [a,b] , c<d$ ; then $f$ is identically zero on $[a,b]$?

Let $f:[a,b] \to \mathbb R$ be a continuous function such that $\int_c^d f(x)dx=0 , \forall c,d \in [a,b] , c<d$ ; then is it true that $f(x)=0 , \forall x \in [a,b]$ ?
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3answers
70 views

Proving an inequality between $\frac 1{n+1}$ and $\frac 1n$ and a definite integral

For all natural numbers $n$, prove that $$\frac 1{n+1} < \int_n^{n+1} \frac 1t \, dt < \frac 1n$$ I have tried working with $\frac 1{t+1} < \frac 1t < \frac 1{t-1}$ but this doesn't ...
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1answer
55 views

Finding an integral. [on hold]

Evaluate $$\!\int (x^5\sqrt{x} + x\sqrt[4]{x})\ \mathrm{d}x$$ My attempt: I tried to factor out a $\sqrt{x}$ and I got $$\sqrt{x}\int\! x^5+x\sqrt[3]{x} \ \mathrm{d}x$$ But here I cannot factor a ...
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3answers
149 views

A reason for the value of $\int_{0}^{1}\log{(x)}\log{(1-x)}\,\mathrm{d}x$

In this .pdf document, which is just a list of Putnam-style undergraduate-level problems from various sources, the third question is as I have stated it below (up to a change of notation). ...
2
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0answers
51 views

How to evaluate the integral $\int\frac{1-e^{-2y} -\frac{2}{k}\ln{(1+ky)}}{(1+ky)e^{-2y}-1}dy$ [on hold]

Please help me in doing this integration. $\int_{0}^{m}\frac{1-e^{-2y} -\frac{2}{k}\ln{(1+ky)}}{(1+ky)e^{-2y}-1}dy$ where m is a positive number.
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0answers
44 views

Subdifferential of integral

I am currently trying to extend my knowledge about subdifferentials. Now I am stuck at a particular property of the subdifferential. In this "paper" ...
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2answers
121 views

A singular Gronwall inequality

Let $f : [0,T] \to R^+$ be a continuous function such that $f(0)=0 $ and : $$ f(t)\le C\int_0^t s^{-1}f(s) ds,\; \forall t\in [0,T] $$ for some constant $C>0.$ Is it true that $f(t)=0,\; \forall ...
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2answers
48 views

Calculate double integral $\iint_A \sin (x+y) dxdy$

Calculate double integral $$\iint_A \sin (x+y) dxdy$$ where: $$A=\{ \left(x,y \right)\in \mathbb{R}^2: 0 \le x \le \pi, 0 \le y \le \pi\}$$ How to calculate that? $x+y$ in sin is confusing as i do not ...
3
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2answers
40 views

Double integral $\int\int_A y dx dy$

Calculate Double integral $$\iint_A y dxdy$$ where: $$A=\{(x,y)\in\mathbb{R}^2 : x^2+y^2\le4, y \ge 0 \}$$ I do not know what would be the limit of integration if i change this to polar coordinates. ...
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1answer
75 views

Why is it incorrect to integrate by $d(2x)$?

I tried to prove the volume of a cone. If you let the radius be $r$ and let the height be equal to the radius, then all you need to do is integrate the area of a circle with radius $r$ by $dr$. ...
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2answers
138 views

Solve this integral:$\int_0^\infty\dfrac{\arctan x}{x(x^2+1)}\mathrm dx$

I occasionally found that $\displaystyle\int_0^{\frac{\pi}{2}}\dfrac{x}{\tan x}=\dfrac{\pi}{2}\ln 2$. I tried that $$\int_0^{\frac{\pi}{2}}\dfrac{x}{\tan x}=\int_0^{\frac{\pi}{2}}x \ \mathrm ...
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0answers
32 views

A little hard double integral

$\iint \frac{2x^2e^{x^2}}{x^2+y^2}dxdy\::\:D=\left\{1\le x\le 2,\:0\le y\le x\right\}$ I use the substitution: $u=x^2,\:v=\frac{y}{x}$ $$$$Then I get: ...
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0answers
27 views

How does one integrate a function where the numerator is a polynomial of a degree n, and the denominator is a polynomial under root of degree m<n?

How does one integrate a function where the numerator is a polynomial of degree $n$, and the denominator is a polynomial under root of degree $m$ $(m<n)$? A random example being ...
1
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1answer
39 views

Calculating $\iint_{D} \left(x-y\right)dxdy$ where $D=\left\{0\le x-y\le 1,\:1\le xy\le 2\right\}$

$$\iint_{D} \left(x-y\right)dxdy$$ where $D=\left\{0\le x-y\le 1,\:1\le xy\le 2\right\}$ So the substitution is pretty obvious, but j is: $J\:=\frac{1}{x+y}$ $$$$ I dont see how I get rid of the ...
0
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1answer
21 views

Problem: conservative and not conservative $F=\left( y+\frac{y}{x^2+y^2}, x-\frac{x}{x^2+y^2}\right)$

I don't know how I can solve this problem: Consider $$F=\left( y+\frac{y}{x^2+y^2}, x-\frac{x}{x^2+y^2}\right).$$ Proving that $F$ is not conservative in $\mathbb{R}^2-(0,0)$ but is conservative ...
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3answers
65 views

How to prove that this function is integrable on $[0,1]$

Here I tried to find two step functions, one of them is less than $f$ on $[0,1]$ whereas one of them is greater than $f$ on the same closed interval, to prove this function is Riemann-integrable on ...
2
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1answer
70 views

Is there a way of solving integrals where the numerator is an integral of the denominator?

Is there a way of solving integrals where the numerator is an integral of the denominator? I was evaluating the integral $$\int \frac{x-\sin x}{1-\cos x}\mathrm{d}x$$. I separated the numerator into ...
1
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1answer
29 views

How does this integration by substitution work?

We have an elliptic area defined by $$A := \{(x, y) \in \mathbb{R}^2 \mid (\tfrac{x}{a})^2+(\tfrac{y}{b})^2 \leq 1 \}$$ and a height function $$h \colon \mathbb{R}^2 \to \mathbb{R}, (x, y) \mapsto ...
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1answer
39 views

Partial integration of $\sin x\log(y-1)$ w.r.t. $x$

If I have the function $\sin x\log(y-1)$ and I want to partially integrate it w.r.t. $x$ then what happens to $\log $? Would the solution be: $-\cos x \log(y-1)$ and how? Isn't $\log(y-1)$ a function ...
0
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1answer
19 views

Trouble with integration in order to find analytic function

Let $u(x,y) = x/(x^2 - y^2)$ Find $v(x,y)$ such that $f(z) = u + iv$ I'm applying Cauchy-Riemann $u_x = -\frac{(x^2 - y^2)}{(x^2 + y^2)} = v_y$ But I don't see how to integrate that with respect ...
0
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1answer
15 views

Abel inversion where axisymmetric function is multiplied by $\sin(\phi)$

I have a problem seems similar to Abel inversion, but the axisymmetric function is multiplied by $\cos{\phi}$, making the integrand non-axisymmetric. Here is a picture of the problem: Each chord is ...
0
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0answers
23 views

Restriction over pdf such that an integral inequality holds $\int_{-\infty}^{+\infty}\left(F(x)-\frac{2}{3}\right)xf(x)dx\geq 0$

Let $f(x)$ be a pdf in $(-\infty,+\infty)$ and $F(x)$ it's cdf. Assume both are smooth. I need to find restrictions over the pdf such that the following inequality holds: ...
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0answers
32 views

Solving an integral that includes an exponential function and the error function

This question contains all the values needed to compute an equation. My question is, do you get the same result I get? Or do you get the result in the paper I've linked to? I'm trying to decipher ...
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2answers
66 views

Taylor Series for $\frac{1}{ 1+x+x^2}$

I tried to solve it in a way. The solution did not match. Please tell me where i went wrong. $\cfrac {1} {1+x+x^2} = \cfrac 4 {4+4x+ 4x^2} = \cfrac 4{ 3+(2x+1)^2} = \cfrac 1{\sqrt 3}\cdot\cfrac 4{ 1+ ...
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0answers
21 views

How to obtain accumulated counts of past events by time $t$?

Given $f: [0, \infty) \to \{0,1\}$, $f(t)$ represents whether there is an event occurring at time $t$. How can we obtain $g: [0,\infty) \to \mathbb{N}_0$ so that $g(t)$ represents the number of ...
1
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4answers
42 views

Integration of the square root of a quadratic

I am in the tricky situation of trying to integrate the following. $$\sqrt{4 a^2 (y-b)^2+c^4}$$ $a, b$ and $c$ are all known constants. Can anybody provide insight as to how to do this? I have ...
2
votes
2answers
74 views

Result of $\int \limits_{-\infty}^{+\infty}x^2\times\exp\left(\dfrac{-x^2}{2}\right)\mathrm{d}x$ [duplicate]

I would like to read a very thorough and explained calculation process for a couple of integrals. For the life of me I just can't figure out the result on my own, and no resource on the web were able ...
4
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0answers
79 views

Another integral related to Fresnel integrals

How would we prove this result by real methods ? $$\int_0^{\infty } \frac{\sin \left(\pi x^2\right)}{x+2} \, dx=\frac{1}{4} \left(\pi-2 \pi C\left(2 \sqrt{2}\right)-2 \pi S\left(2 ...
1
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4answers
50 views

Integrating linear/trigonometric

I have the following question- $\int$ $\frac{x}{1+cosx}dx$ Do I do integration by parts or is there some other method? Thanks for the help.
2
votes
2answers
28 views

Integration of a scalar function with respect to a vector

I have a scalar function that takes $n$ arguments, $f(x_1, x_2,x_n) = f(\mathbf{x})$, and two vectors also with $n$ elements, $\mathbf{z} = (z_1, z_2\cdots,, z_n)$, and $\Delta\mathbf{z} = (\Delta ...