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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2
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0answers
9 views

How to reverse the integration order of the double integral $\int_{\theta=0}^{2\pi}\int_{r=0}^{1+\cos\theta}r^2(\sin\theta+\cos\theta)drd\theta$.

I am given the integral $$ \iint\limits_H \, (x+y) \mathrm{d} A $$ where $H$ is the area of the cardioid $r=cos(\theta)+1$. I have translated the double integral to polar coordinates in order to solve ...
1
vote
1answer
45 views

How do I solve $\int_ {-\infty}^\infty xe^{-m(x-a)^2}$?

Evaluate $$\int_{-\infty}^\infty xe^{-m(x-a)^2}$$ where $m$ and $a$ are constants I can solve this if the exponential is simple $e^{x^2}$ by substitution, but this one doesn't work that way as ...
0
votes
1answer
12 views
-6
votes
1answer
27 views

Integral: $\int \frac{x - 4}{\sqrt{x^2-2}}dx $ [on hold]

$$\int \frac{x - 4}{\sqrt{x^2-2}}dx $$
0
votes
1answer
21 views

For which $a$ does the integral $\int_B||x||^{-a}dx$ exist, where $B:= \{ x\in \mathbb{R^2} :||x|| \leq1\} $?

For which $a$ does the integral $\int_B||x||^{-a}dx$ exist. $B:= \{ x\in \mathbb{R^2} :||x|| \leq1\} $ My solution was transforming into polar coordinates, $x=\cos(\phi)*r,y=\sin(\phi)*r$ such that ...
0
votes
0answers
68 views

Are there chain rules for integration possible?

In "A Quotient Rule Integration by Parts Formula" and in "Quotient-Rule-Integration-by-Parts", the authors integrate the product rule of differentiation and the quotient rule of differentiation and ...
0
votes
3answers
46 views

Integration of $(ae)^x$

I just found that $$\int a^x\ \text{d}x = \frac{a^x}{\log(a)}$$ $$\int e^x\ \text{d}x = e^x$$ But what about $$\int(a e)^x\ \text{d}x$$ I guessed it should be $\frac{(ae)^x}{\log(a)}$ But I am ...
5
votes
4answers
78 views

Evaluate the integral $\int^{\infty}_{0} e^{-x}x^{100}dx$

$$\int^{\infty}_{0} e^{-x}x^{100}dx$$ I am sure is something here I can not see, else it is integration by parts 100 times.
-2
votes
1answer
50 views

evalute $\int \frac{1}{x^6+x^4+7x^3+7x}\mathrm{ d}x$ [on hold]

I want to evalute this integral $$\int \frac{1}{x^6+x^4+7x^3+7x}\mathrm{ d}x$$
1
vote
1answer
26 views

If $H$ is the area of the cardioid $r=1+\cos( \theta )$, calculate the double integral $ \iint\limits_H \, (x+y) \mathrm{d} A. $

As stated in the title, I'm trying to calculate $$ \iint\limits_H \, (x+y) \mathrm{d} A $$ Where $H$ is the area of the cardioid $r=1+\cos (\theta) $. I have calculated the area of $H$ to be $$ ...
0
votes
0answers
36 views

How can show that the following function is non-negative?

I was working on a problem and reduced it to show the following inequality: ‎‎ $$\sum_{\substack{i,j=1\\i<j}}^{n}x_i^{\alpha}x_j^{\alpha}(\ln x_i-\ln x_j)^2+A_1\Big[\sum_{i=1}^{n}x_i^\alpha (\ln ...
1
vote
1answer
40 views

How to solve this double integral $\iint_{D}\frac{1}{x+6}dxdy$

How can I solve this double integral? $$\iint_{D}\frac{1}{x+6}dxdy$$ Where $D$ is the region between $y-axis$ and the parametric curve $$ \left\{\begin{matrix} x(t)=t-t^3\\ y(t)=2t-t^2 ...
1
vote
2answers
54 views

Evaluation of $\int_{0}^{1}\frac{\arctan x}{1+x}dx$

Evaluation of $$\int_{0}^{1}\frac{\tan^{-1}(x)}{1+x}dx = \int_{0}^{1}\frac{\arctan x}{1+x}dx$$ $\bf{My\; Try::}$ Let $$I = \int_{0}^{1}\frac{\tan^{-1}(ax)}{1+x}dx$$ Then $$\frac{dI}{da} = ...
0
votes
0answers
15 views

Evaluating an integral (real part, cauchy pdf)

I was trying to find a characteristic function for a cauchy pdf, without going into contour integration (because I have no idea how to do it). I have: $$\hat{f}(\xi)=\frac{\gamma}{\pi}\int_{\Bbb R} ...
0
votes
0answers
6 views

Surface Integral of a harmonic function and mean value property

I want to find a general expression of the following integral, where $h$ is a harmonic function (we're in $\mathbb{R}^2$): $\int_{|x-y|\leq a^2}\frac{h(y)}{\sqrt{a^2-|x-y|^2}}dy$ I think I can ...
2
votes
1answer
38 views

Integration involving greatest integer function : $\int_0^{\pi} [cot(x)]dx$

What the integral of $$\int_0^{\pi} [\cot(x)]dx$$ where $[\cdot]$ represents greatest integer function. I know integral of $\cot$ is $|\log(\sin(x))|$ but $\log$ is not defined for $0$ or is there ...
0
votes
1answer
15 views

$g(x) \ge 0$ Riemann integrable on $[a,b]$ then for each subinterval $\int^b_a g(x)dx \ge \int^d_c g(x)dx$

Let $g(x) \ge 0$ Riemann integrable on $[a,b]$. Show using Riemann sums that for each subinterval $[c,d] \subset [a,b]$: $$\int^b_a g(x)dx \ge \int^d_c g(x)dx$$ I thought that we should ...
0
votes
1answer
41 views

Definite integral with cube roots of trig functions [on hold]

Find a closed form for the following integral: $$\int _{\pi/6} ^{\pi /3} \frac {\sqrt[3]{\sin x}}{\sqrt [3]{\sin x} + \sqrt[3]{\cos x}}dx$$ I think the answer is found out using some properties. So ...
1
vote
1answer
22 views

Computation of an integral depending on the Legendre polynomials

Let $P_l$ be a Legendre polynomial ($l$ is an integer). I want to know why the quantity $$ v_l(k):=(-i)^l\int_{-1}^{+1}\mathrm{e}^{ikx}\,P_l(x)\;\mathrm{d}x $$ is real?
0
votes
0answers
14 views

Construction of the Area Function

I am following calculus by Tom M Apostol in which he has given the Axiomatic definition of the Area Function We assume there exists a class M of measurable sets in the plane and a set function a, ...
2
votes
1answer
21 views

Change of variables, integration

In a finite element analysis, I am evaluating the following integral: $$\int_{0}^{h}\left ( 1-\frac{x}{h} \right )*\left ( x \right )dx$$ but I want to apply a transformation from x to integrate ...
22
votes
9answers
1k views

Is there any integral for the Golden Ratio?

This is a curiosity. I was wondering about math important/famous constants, like $e$, $\pi$, $\gamma$ and obviously $\phi$. The first three ones are really well known, and there are lots of integrals ...
0
votes
1answer
52 views

How to find the center of mass in this problem

How can I find the centre of mass of the surface of the sphere $x^2+y^2+z^2=a^2$ that is contained in the cone $z\tan(\gamma)=\sqrt{x^2+y^2}$, $0 \lt \gamma \lt$ $\pi/2$ a constant, where the density ...
1
vote
2answers
29 views

Find the original function by using convolution theorem

Seems like I don't know how to apply convolution theorem on this problem properly, I would appreciate some help and a brief explanation how did you solve it if you do it. ...
0
votes
1answer
33 views

If $f(x)=\lim_{n\to\infty}n^2(e^{\frac{k}{n}\ln\sin x}-e^{\frac{k}{n+1}\ln\sin x})$ where $0<x<\pi$, $n\in\mathbb{N}$

If $f(x)=\lim_{n\to\infty}n^2(e^{\frac{k}{n}\ln\sin x}-e^{\frac{k}{n+1}\ln\sin x})$ where $0<x<\pi$, $n\in\mathbb{N}$ and $\int_0^{\frac{\pi}{2}}f(x)dx=-\frac{\pi}{k}\ln4$, then the value of ...
1
vote
1answer
40 views

Integrating inverse trig function with radicals

$$\dfrac{x + 5}{\sqrt{9-(x-3)^2}}$$ It's a inverse trig integration problem. I tried to separate the numerators but made my problem worse. Any advice?
-1
votes
1answer
36 views

Help with vector triple integral problem

Prove that $$\iiint_{D}(\vec a \cdot \vec R)(\vec b \cdot\vec R)(\vec c \cdot\vec R) \,dx\,dy\,dz=\frac{(\alpha\beta\gamma)^2}{8r}$$ Where the $\vec a , \vec b,\vec c$ are constant vectors, $\vec ...
2
votes
2answers
58 views

Integral $\frac{\sin(x)}{x}$ finite domain

I have seen a question asking to find the value of $\int_{-100}^{100} \frac{\sin{x}}{x} dx$. I have to confess that I didn't think this was possible. If I expand the $\sin$ using Taylor series, ...
0
votes
0answers
15 views

Change of variables from unit unit ball to another ball for integration

What is the general formula for changing coordinates for integration from the unit ball to another ball? For example, if I wanted to change from integrating $f(x-r)$ over $B(0,1)$, the open ball about ...
0
votes
3answers
77 views

Integrate $\int{\frac{1}{(x+1)(x+2)^2(x+3)^3}dx}$

How to integrate this $$\int{\frac{1}{(x+1)(x+2)^2(x+3)^3}dx}$$ I tried to use that $$\int{\frac{1}{(x+1)(x+2)^2(x+3)^3}dx} = P_{1}(x)/Q_{1}(x) + \int{P_{2}(x)/Q_{2}(x)dx}$$ where ...
1
vote
0answers
50 views

Indefinite integration and definite integration calculation

Given the functions: $$\begin{cases} B(x)= \dfrac {x\ln(1+\ln(x))}{1+x^{4/3}}\\[2ex] A(x)=B(2x-1)-B(2x) \end{cases}$$ for $b>1$ find out: $$\int_{b}^\infty A(x) dx - \frac12\int_{2b-1}^{2b} B(x) ...
0
votes
1answer
16 views

Trapezoid rule for finding coefficient

If we know that $\int_{a}^b t(x)=h \sum_{k=1}^2 dk * t(a+kh)+O(h^m)$ where $h=\frac{b-a}{3}$, how do we find the coefficient d1, d2 and m in the equation? Answer says that d1=3/2, d2=3/2, m=3 I ...
7
votes
2answers
99 views

Anti-derivative of continuous function $\frac{1}{2+\sin x}$

I use tangent half-angle substitution to calculate this indefinite integral: $$ \int \frac{1}{2+\sin x}\,dx = \frac{2}{\sqrt{3}}\tan^{-1}\frac{2\tan \frac{x}{2}+1}{\sqrt{3}}+\text{constant}. $$ ...
0
votes
2answers
28 views

Integral of a measurable function

I do not know what should i keep as title for this question... Question goes like this.. Let $f:\mathbb{R}\rightarrow [0,\infty)$ be a measurable function. If $\int_{-\infty}^{\infty}f(x)dx=1$ prove ...
2
votes
0answers
63 views

difficult integral $\int_0^{\pi/2}\frac{x^2({1+\tan^2 x})^2}{\sqrt{\tan x}({1-\tan x})}\sin{4x}dx$

This is a complicated integral, the numerical value appears to me correct.Therefore how to prove this result?$$I=\int_0^{\pi/2}\frac{x^2({1+\tan^2 x})^2}{\sqrt{\tan x}({1-\tan ...
0
votes
0answers
63 views

Solving an indefinite integral problem [on hold]

The given problem is $$ \int \frac{2^{\sin x}}{2^{\sin x} + 2^{\cos x}} dx$$ please help me solving this indefinite integral problem...thank you very much, actually I have solved an definite integral ...
0
votes
0answers
33 views

Proving an inequality involving integrals?

I am trying to prove that $$[\sum_{i=1}^{n}(\ln t_i)^2 t_i^\alpha+A^{\prime \prime}(\alpha)][\sum_{i=1}^{n}t_i^\alpha+A(\alpha)]\ge[\sum_{i=1}^{n}(\ln t_i) t_i^\alpha+A^{\prime}(\alpha)]^2$$ where ...
0
votes
0answers
50 views

how to evalute this equality

I want to prove this equality $$ \frac{1}{2\pi}\frac{(x-y)\cdot y}{(x_1-y_1)^2+(x_2-y_2)^2}= \frac{ab}{4\pi}\frac{1}{a^2\sin^2(\alpha+\beta)+b^2\cos^2(\alpha+\beta)}.\tag{1}$$ where ...
0
votes
1answer
22 views

Integral, partial fractions, need explanation for how to get from one step to another.

Can someone explain how they go from the red step to the blue one?
0
votes
1answer
25 views

How to Proceed in Solving this Equation

Let $f: [0,\infty)\to \mathbb{R}$ a non-decreasing function. Then show this inequality holds for all $x,y,z$ such that $0\le x<y<z$. \begin{align*} & (z-x)\int_{y}^{z}f(u)\,\mathrm{du}\ge ...
1
vote
1answer
26 views

Proving a reduction formula. $\cos^n (2x)$

Establish a reduction formula for $$\int \cos^n (2x)dx$$ My attempt, Let $I_n=\int \cos^n 2x dx$ $=\int \cos^{n-1}2x (\cos 2x dx)$ Let$$u=\cos^{n-1}2x$$ $$du=-2(n-1)\cos^{n-2}2x (\sin 2x)dx$$ ...
0
votes
0answers
27 views

Bounding an integral

I'm trying to show that the following integral ( a solution for the non-homogeneous transport equation ) has this bound: $$ \begin{equation*} \left\|{ \int_{0}^{t} f(x+b(w-t),w) dw ...
1
vote
2answers
43 views

Sign of the error in Simpson's rule

Let $f : [a,b] \to \mathbb{R}$ be a $C^\infty$ function. The Riemann integral $I = \int_a^b f(x)\,dx$ can be approximated by using Simpson's rule: $$I \approx S = \frac{b-a}{6} \left[ f(a) + 4 ...
4
votes
0answers
45 views

An alternative way to determine when $\int_{0}^{\infty} \cos(\alpha x) \prod_{m=1}^{n} J_{0}(\beta_{m} x) \, dx =0$

Let $J_{0}(z)$ be the Bessel function of the first kind of order zero, and assume that $\alpha$ and $\beta_{m}$ are positive real parameters. When $|z|$ is large in magnitude and $-\pi < \arg(z) ...
1
vote
1answer
67 views

Integrate $\int{ \left( \frac{1-x}{1+x} \right)^\frac{3}{2}dx}$

Integrate $$\int{ \left(\frac{1-x}{1+x} \right)^\frac{3}{2}dx}$$ I guess that there is sub $x = \cos t$ so integral gets to $$\int{ \left(\tan \frac{t}{2} \right)^3 d\cos t}$$ then I used that $\sin t ...
0
votes
1answer
25 views

Surface are of a curve $y=\sin \left(\frac{\pi x}{6} \right)$ rotated about the $x$ axis.

I'm doing a problem involving finding the surface area of the curve for $y=\sin \left(\frac{\pi x}{6} \right)$, rotated about the $x$ axis, for $[0 < x < 6]$. I got as far as $\frac{72}{\pi} ...
0
votes
1answer
23 views

Problem with convolution, insecure

$$f(t)= t^2\cdot u(t),\quad g(t)=t^4\cdot u(t)$$ I know that I need to use convolution theorem to solve this problem, but I really don't know what to do with step functions. Do I need to include ...
0
votes
1answer
39 views

Evaluating this surface integral

I want to evaluate the following surface integral $$\int_S d S \, \, \exp\left(\frac{\Gamma^2 (x^2 \sigma_x^2 + y^2 \sigma_y^2 + z^2 \sigma_z^2) - 2 c R \Gamma (x x_0 + y y_0 + z z_0)}{2 c^2 ...
2
votes
4answers
72 views

Clever way of calculating the integral $ \int \frac{dt}{t^2\sqrt{t-2} } $

$$ \int \frac{\text{d}t}{t^2\sqrt{t-2} } $$ I know it can be calculated using somewhat complicated substitutions, but is there possibly some clever way of solving that type of integral? I don't ...
1
vote
0answers
17 views

Integral of least squares and general rules of integration to solve the integral.

My calculus is very rusty and I am interested to know if the following is solvable: $$ \int_0^{\pi}( \log( \frac {(x_0 + e^{-i\omega})(x_0 + e^{i\omega})(x_1 + ...