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
31 views

Finding an integral.

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
2answers
103 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
38 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. $\int\frac{1-e^{-2y} -\frac{2}{k}\ln{(1+ky)}}{(1+ky)e^{-2y}-1}dy$
3
votes
0answers
35 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
110 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 ...
0
votes
0answers
25 views

Are Lebesgue's Monotone Convergence Theorem and Dominated Convergence Theorem the only case for taking the limit inside the integration?

What are all the possible cases in which we can take limit inside the integration sign? Monotone Convergence Theorem and Dominated Convergence Theorem gives us some conditions for taking the limit ...
1
vote
2answers
39 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
36 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
73 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
1answer
96 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
38 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
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 ...
0
votes
3answers
63 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
votes
1answer
65 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
vote
1answer
25 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 ...
0
votes
1answer
37 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
votes
1answer
17 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
votes
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
votes
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: ...
0
votes
0answers
30 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 ...
-1
votes
2answers
64 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+ ...
0
votes
0answers
20 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
vote
4answers
41 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
72 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
votes
0answers
77 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
vote
4answers
45 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
27 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 ...
-5
votes
0answers
64 views

Hardest integration ever? [on hold]

integrate $\int \frac{1}{1+x^4 }dx$ I have no idea how to do this, I have a test tomorrow pls help! I tried adding and subtracting $2x^2$ and use $a^2-b^2$.
1
vote
1answer
97 views

$\int_0^1(1+\log(x))\sin(x)dx$ How to solve this Integral?

$$\int\limits_0^1(1+\log(x))\sin(x)dx$$ Someone has challenged me to solve this, I solved it without bounds, I have no idea how to do it with those limits.. Is $u=1+\log(x)$ right substituion? or ...
0
votes
0answers
30 views

Differentiating CDF

I'm trying to differentiate the cdf of z with respect to x where the upper bound is a function of x and z ~ N(a , $b^2$ $\cdot$ $x^{-2}$) $\frac{d}{dx} \int _{-\infty} ^ {z^*(x)} \Phi ^{\prime} (z) ...
2
votes
2answers
107 views

How to solve $ \int \limits _0 ^{2\pi} \dfrac{dx}{(\alpha +\beta\cos x)^2} $

I am trying to solve this integral, I think that it could be solve using the complex. $$ \int \limits _0 ^{2\pi} \dfrac{dx}{(\alpha +\beta\cos x)^2} $$
0
votes
1answer
40 views

Differentiation Commute with Lebesgue Integration

My question is simple: Given $f: \mathbb{R}^{n+m} \to \mathbb{R}$, $f\in C^{k}(\mathbb{R}^{n+m})$ , and $X \subset \mathbb{R}^{n}$. Write $f$ as $f(x_1, \ldots, x_n, t_1, \ldots t_m)$. When is ...
0
votes
3answers
50 views

Intuitive meaning of the probability density function at a point

I understand how to integrate probability density functions to find probability within a certain range. However, what I don't understand is what it would mean to set the variable (say x or y) to a ...
0
votes
0answers
35 views

Is it possible for the Simpson's method to converge faster than Rombergs method?

I have the following integral: $\int_{0}^{100} \frac{x^{3/2}}{\cosh{(x)}}dx$ I am running code for the Simpson's method and Romberg method to evaluate the integral numerically and the results show ...
2
votes
3answers
95 views

Primitive of $\int { \frac { x^{ 2 } }{ (x\sin x+\cos x)^{ 2 } } dx } $

How do I evaluate the integral of $$\int { \frac { x^{ 2 } }{ (x\sin x+\cos x)^{ 2 } } dx } $$ in a simple way? The way I could do the question, was by multiplying and dividing the fraction by $\cos ...
3
votes
3answers
59 views

Reduction formulae in definite integration

$$I_n = \int_0^{\pi}\frac{\sin^2(nx)}{\sin^2(x)}dx $$ Find relation between $I_n$, $I_{n+1}$ and $I_{n+2}$ I tried integration by parts by taking $\sin^2(nx)$ as the first function, but reached ...
3
votes
1answer
98 views

Calculating $\int_0^{\pi/4} \frac{\cot (x)}{\cot ^2(x)+\sqrt{\cot (x)}} \, dx$

This is not really one of that kind of integrals that Mathematica cannot handle with, but given the case of a contest, how would we like to handle with it? I would like so much to know your ideas ...
0
votes
0answers
26 views

Integral of exponential complex trigonometric functions

I have a problem with this integral: $X_{e11}$ = $\int_{0}^{2\pi} \int_{0}^{\pi}e^{-ikr_{n}\left(\sin\vartheta \cos \varphi \sin \theta_n \cos \phi_n + \sin \vartheta \cos \varphi \sin \theta_n \sin ...
0
votes
2answers
30 views

Why is the estimate of the order of error in Trapezoid converging to $2.5$?

The integral in question is: $\int_{0}^{\infty} \frac{x^{3/2}}{\cosh{(x)}}dx$ I coded a program to compute $p$, an estimate of the order of the error for the Trapezoid method of numerical ...
0
votes
1answer
20 views

Evaluating This Complex Line integral

I'm trying to evaluate the following: $$\int_{\mathcal{C}}z^3 e^{-z^4}\,dz $$ along the path $\mathcal{C}=\left\{\sin(t^2)-i\frac{2t^2}{\pi}:0\leq t\leq\sqrt{\frac{\pi}{2}}\right\}.$ I tried using ...
1
vote
1answer
50 views

Definite Integration [on hold]

For some constant c, we wish to compute the following integration (or a tighter bound on the same) $\int_{\theta}^{1} x \exp \left(- \frac{c\theta}{x}\right) dx $
1
vote
1answer
67 views

Closed form of this sum

$$\sum _{ s=1 }^{ \infty }{ \left( \frac { 1 }{ 4s-1 } \sum _{ n=0 }^{ \infty }{ \left( \frac { 1 }{ n+1 } \sum _{ k=0 }^{ n }{ \left( \left( \begin{matrix} n \\ k \end{matrix} \right) \frac { { ...
1
vote
0answers
36 views

Integrals with error function and exponentials

I'm trying to solve the integrals below: $$\int_{-\infty}^\infty \int_{-\infty}^\infty \frac{x}{\sqrt{x^2+y^2}}\cdot \operatorname{erf}\left(m\cdot\sqrt{x^2+y^2}\right) \cdot \exp(-a\cdot ...
1
vote
3answers
51 views

Limits using definite integration

$F(k)$ = $$ \lim_{n\to \infty}{\frac{1^k + 2^k +...+n^k}{(1^2 + 2^2 +...+n^2)*(1^3 + 2^3 +...+n^3)}} $$ I need help in finding $F(5)$ and $F(6)$. I tried converting it into summation form and using ...
4
votes
2answers
77 views

Evaluating definite integral of $e^{i t^2}$

In passing Sakurai's QM book mentions that $$\int_{-\infty}^\infty e^{i t^2} dt = \sqrt{i \pi}$$ This is consistent with 7.4.4 in Abramowitz and Stegun which claims for $\Re a > 0, n = 0, 1, 2, ...
-3
votes
0answers
32 views

Conditions for Riemann integrability [on hold]

A function $f$ is Riemann-Integrable iff the infimum of the upper sum and the supremum of the lower sum of all partitions P of a closed interval [a,b] in the domain of $f$ coincide, as stated below: ...
0
votes
0answers
29 views

Solving for the limit of a Gaussian random variable within an integral

I'm having trouble solving a particular integral. It is $$ (1/\Delta t)\int_t^{t+\Delta t}I(t')dt', $$ where $$ I(t') = \mu_c+\sigma_c \eta(t'). $$ In this second equation, $$ \eta(t') = ...
1
vote
1answer
27 views

Integration divided by the function

How do I guarantee that $ \frac{\int_0^v f(x) dx}{f(v)} $ is increasing? Under which assumptions is this true? Or, what types of properties would such a function have? Thanks.