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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1answer
29 views

Can I solve this integral with a squared sum in it?

Title says it all. By now I have tried by hand and I think that it is indeed solvable, but I can't handle the very long terms. I tried to run the thing through SAGEs integrator: ...
1
vote
0answers
10 views

Solve complex integral with $\Gamma$-function

Let $s\in\mathbb C$ and $r\in\mathbb R$. In the integral $$\int_{-\infty}^\infty \frac{1}{z^{r+s}\overline{z}^s} dx$$ we have $z=x+iy$ where $y>0$ is fixed. I read that you can explicitly compute ...
0
votes
3answers
39 views

Tricky Integration And Functions Question

If there is a functions $f(x)$ such that $$ f(x) = x+\int_0^{\frac{\pi}{2}} \sin(x+y)\cdot f(y) \, dy $$ I tried doing it but it seems to get more and more complex as I proceed. Find $f(x)$ Thanks
-8
votes
1answer
41 views

Can somebody integrate this function for me? [on hold]

This is the function. $\frac{1}{6.08 \cdot \sqrt{2\pi}}\exp\left(-\frac{(x-10.75)^2}{2 \cdot 6.08^2}\right)$ Thanks in advance!
10
votes
0answers
88 views

A very tough integral $\int_0^{\pi} \arctan^3\left(\frac{\sin (x)}{2 \sqrt{2}}\right)\csc ( x) \, dx$

My research shows that $$\int_0^{\pi} \arctan^3\left(\frac{\sin (x)}{2 \sqrt{2}}\right)\csc ( x) \, dx$$ $$=\frac{3}{16} \pi \sinh ^{-1}(1) \log ^2(2)-\frac{1}{96} 85 \pi \log ^3(2)+\frac{61}{16} ...
0
votes
0answers
18 views

Double integral over a triangle

Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ be a smooth function (derivable, integrable over all of $\mathbb{R}^2$). Let $T$ be a triangle in $\mathbb{R}^2$, defined by its vertices : $A=(x_a,y_a)$, ...
0
votes
0answers
17 views

Upper bound for incomlete Gamma function

It is well-known, that for real arguments $a \geq 0$ and $x \geq 0$ the upper incomplete Gamma function $$\Gamma(a,x) = \int_x^\infty e^{-t} t^{a-1} \, \mathrm{d} t$$ behaves for sufficiently large ...
0
votes
0answers
25 views

Prove that $\int_c^d{f(y)dy} = \int_a^b{f(G(x))dG(x)}$

I'm doing this exercise from Real Analysis of Folland and got stuck on this problem. Let $G$ be a continuous increasing function on $[a, b]$ and let $G(a) = c, G(b) = d$. a) If $E ...
0
votes
3answers
34 views

fundamental theorem of calculus 2 [on hold]

Differentiate the following equation with respect to $x$: $$8 + \int_a^x \frac{f(t)}{t^2}\, dt = 2 x^{1/2}$$ Hence, find a function $f(x)$ and real number $a$ such that the above equation is true ...
0
votes
0answers
13 views

$f:[a,b] \to [0, \infty)$ continuous , then $\lim_{n \to \infty} \Bigg(\int_a^b \big(f(x)\big)^ndx \Bigg)^{1/n}=\sup \{f(x):x \in [a,b]\}$ ? [duplicate]

Let $f:[a,b] \to [0, \infty)$ be continuous , then is it true that $\lim_{n \to \infty} \Bigg(\int_a^b \big(f(x)\big)^ndx \Bigg)^{1/n}=\sup \{f(x):x \in [a,b]\}$ ?
-1
votes
0answers
29 views

When we take integration of any function, then what exactly we do with it? [on hold]

Ex. $\int 2x\, dx= x^2$Then what we have exactly done with function.
3
votes
0answers
15 views

Correct bounds for simple triple integral in rectangular coordinates?

This is homework, so I am not after a solution to this problem. I am required to evaluate the integral $\iiint_{V}y\;dV$. $V$ here is the solid bounded above by the plane $x+y+z=1$ and by the ...
1
vote
1answer
43 views

what is difference between summation and integration? explain with example. [on hold]

I want to distinguish between obtaining process of integration and summation.I.e what we exactly do when we take summation or integration of any function.
0
votes
1answer
37 views

How do we calculate the upper sum and lower sum of an Integral?

How do we calculate the Upper and Lower Sum of an Integral? I am trying to calculate it to for : $$\int_1^2 (3-4x) dx$$ Is there a Formula?
0
votes
0answers
21 views

Demonstrate the convergence of an integral

Can anyone demonstrate that the following integral is convergent? $$\int_0^{\tau"}\left(\int_0^{\tau'}\frac {1}{\tau^2|\ln\tau|^p}d\tau\right)d\tau'$$ $p$ is a constant $>12$.
0
votes
0answers
11 views

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 ...
0
votes
0answers
15 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 ...
1
vote
1answer
48 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 ...
2
votes
3answers
86 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 ...
0
votes
2answers
29 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.
0
votes
0answers
38 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 ...
0
votes
0answers
13 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\ ...
0
votes
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. ...
0
votes
1answer
20 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
votes
1answer
79 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 ...
5
votes
0answers
34 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 ...
3
votes
0answers
61 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
votes
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
votes
1answer
82 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 ...
0
votes
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 ...
1
vote
0answers
16 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
votes
4answers
47 views

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

I need a hint how to start solving this integral: $$\int{ \frac{1}{\sqrt {1 - e^{2x}} } dx}$$
0
votes
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
vote
1answer
34 views

Approximaing Gamma function

For $c>1$ and $0<\theta<1$, we wish to approximate (upper bound) following Gamma function: $$\int_c^{c\theta}x^{-3}e^{-x}dx $$
1
vote
0answers
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} ...
0
votes
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 $f(x)=0 , \forall x \in [a,b]$ ?
3
votes
3answers
74 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 ...
0
votes
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 ...
7
votes
3answers
153 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
votes
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.
3
votes
0answers
46 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" ...
4
votes
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 ...
1
vote
2answers
49 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
votes
2answers
42 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. ...
0
votes
1answer
76 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 $dr$. ...
1
vote
2answers
140 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 ...
0
votes
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: ...
0
votes
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
vote
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
votes
1answer
22 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 ...