Questions tagged [integration]

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 the integrals being considered. This tag often goes along with the (calculus) tag.

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3
votes
2answers
142 views

Integral $\int^{\infty}_0 \exp\left[-\left(4x+\dfrac{9}{x}\right)\right] \sqrt{x}dx$

How do I evaluate $$\displaystyle\int^{\infty}_0 \exp\left[-\left(4x+\dfrac{9}{x}\right)\right] \sqrt{x}\;dx?$$ To my knowledge the following integral should be related to the Gamma function. I ...
5
votes
1answer
67 views

Need help evaluating this improper double integral

Need help evaluating this improper integral $$\iint_D (1-x+y)e^{(-2x+y)}\,dx\,dy$$ where $D$ is the region: $0 \le y \le x $. I've tried doing the substitution: $u = -2x +y \\ v = 1-x+y$ which ...
-1
votes
0answers
34 views

Find $\int (z^4+4z)e^z \cos^2 z \,\mathrm dz$ [closed]

$$\int (z^4+4z)e^z \cos^2 z \,\mathrm{d}z$$ To find this complex integration over the circle $|z-2|=5$
1
vote
0answers
53 views

How can I calculate the surface area of an umbrella? [closed]

Umbrella shape I want to find the surface area of a beach umbrella, similar to what is shown in the image below. I recognize that it is made by intersections of half-cylinders. however, I still can't ...
0
votes
2answers
43 views

What would be the step by step solution of this double integral by changing it to Polar coordinates?

$$\int_0 ^ {1} \int_{-\sqrt{x-x^2}} ^ {\sqrt {x-x^2}} (x^2+y^2) ~dy~dx$$ My findings are: $$\int_?^? \int_?^? r^3 ~dr~d\theta$$ Region is the circle of radius $~\frac{1}{2}~$ centered at $~(\frac{...
0
votes
1answer
33 views

How can I solve $\int x\exp(-Tx^a)\,dx$?

How can I solve $$\int x\exp(-T x^a)\,dx$$ ($T$ and $a$ are variables.) In WolframAlpha, the answer is $$-x^2(Tx^a)^{-2/a}\,\frac{\Gamma(2/a,Tx^a)}a$$ I don't know why.
0
votes
1answer
54 views

Evaluation of some integrals

I have to calculate some double integrals, as: $$ \iint_{D} x^2y dxdy $$ where $D$ is the bounded region between $C={[(x,y)\in R^2:x^2+y^2=4,y\ge0]}$ and $x$-axis. I am not sure about the domain $D$,...
6
votes
1answer
145 views

Mind-blowing Sums: Compute $\sum_{n=1}^\infty\frac{H_nH_n^{(2)}}{n^22^n}$ and $\sum_{n=1}^\infty\frac{H_n^3}{n^22^n}$

How to prove the following two sums \begin{align} \sum_{n=1}^\infty\frac{H_nH_n^{(2)}}{n^22^n}&=2\operatorname{Li}_5\left(\frac12\right)+\ln2\operatorname{Li}_4\left(\frac12\right)-\frac{31}{32}...
1
vote
2answers
78 views

Is $f(y)=\chi\!\left(\bigcup\limits_{k=1}^{\infty}\left[q_{k},q_{k}+\frac{1}{4^{k}}\right]\right)(y)$ Riemann-Integrable on $\mathbb{R}$?

Is $$f(y)=\chi\!\left(\bigcup\limits_{k=1}^{\infty}\left[q_{k},q_{k}+\frac{1}{4^{k}}\right]\right)(y), \hspace{0.2cm} \{q_{k}\}_{k \in \mathbb{N}} \subseteq \mathbb{Q}$$ Riemann-Integrable ? ...
7
votes
1answer
83 views

How to compute $\partial \frac{1}{z^*}$?

I have trouble understanding some basic concepts in Complex Analysis: For $z=x+\mathrm{i}y$, we define: $$\partial \equiv \frac{\partial}{\partial z}=\frac{1}{2}\left(\frac{\partial}{\partial x}-i \...
1
vote
1answer
47 views

Divergence theorem, integrals don't give the same result.

I am trying to find the flux going out of the semi sphere $$z=\sqrt{4-x^2-y^2}$$ using the divergence theorem, but I'm getting different results between the integrals: $$\iiint \operatorname{div} F\, ...
0
votes
0answers
34 views

How to write this system of non-linear differential equations into a system of first order differential equations?

This is the system in question: $m*{x}''+(k+\sigma _{2})*{x}'=\sigma _1*{z}'+ \sigma _0*z+\sigma _2*v_{0}$ ${z}'=(v_{0}-{x}')-\frac{\sigma _0*\left | v_{0}-{x}' \right |}{g(v_{0}-{x}')}*z$ $g(v_{0}-...
0
votes
1answer
43 views

Help with integration of $(f+ig)(f''+ig'')=3(f'+ig')^2$

I have the differential equation $$(f+ig)(f''+ig'')=3(f'+ig')^2,$$ and the solution says that by integrating once you get $$f'+ig'=C(f+ig)^3,$$ and twice $(f+ig)^{−2}=D−Cu.$ I tried with partial ...
8
votes
1answer
102 views

Generalizing Archimedes' “The Quadrature of the Parabola”

In the third century BC Archimedes discovered that The area enclosed by a parabola and a line (left figure) is 4/3 that of a related inscribed triangle (right figure). Consequentially, the area ...
3
votes
1answer
94 views

Find $\int_0^{100\pi}\sqrt{1-\cos2x}dx$

Find $$\int_0^{100\pi}\sqrt{1-\cos2x}dx$$ My work: $$\int\sqrt{1-\cos2x}dx=\sqrt2\int\sqrt{\sin^2x}dx$$ I use wolfram What will I do next?
1
vote
5answers
86 views

Finding the integral $\int \frac{dx}{(1-x)^2\sqrt{1-x^2}}$

One may take $x= \cos t$ and get $$I=\int \frac{dx}{(1-x)^2\sqrt{1-x^2}}= -\frac{1}{4}\int \csc^4(t/2)~ dt=-\frac{1}{4} \int [\csc^2(t/2) +\csc^2(t/2) \cot^2(t/2)]~ dt.$$ $$\Rightarrow I=\frac{1}{2} \...
2
votes
2answers
39 views

What is the surface area of given surface .

Compute the area of that part of plane $x+y+z=2a$ which lies in the first octant and is bounded by the cylinder $x^2+ y^2 =a^2$ Now $z = 2a -x -y$ , $\dfrac{\partial{z}}{\partial{x}} = -1$ $\...
0
votes
0answers
23 views

Definite integrals involving the product of arbitrary Bessel function and arbitrary power of x

Is there a general formula for the following definite integral: $\int^{a}_{b}J_m(\lambda_{mi}x)x^ndx=?$ Where n and m are arbitrary natural numbers and $\lambda_{mi}$ is i-th zero of m-th Bessel ...
1
vote
0answers
33 views

Shell Integration

Define y=$e^x cos(x)$ over the interval [0, π]. Consider the area of the region R bounded by the curve y=$e^x cos(x)$, the lines $x=π/2$, $x=π$ and the x-axis. I want to find the volume generated when ...
0
votes
0answers
59 views

Integration by substitution which involves trigonometry [closed]

Need help on how to do this integration by substitution. $$\int (\sqrt{\sin x} + \sqrt{\cos x}) dx$$
0
votes
0answers
44 views

Fourier Series for a Dirac Train

I'm trying to find out by myself the Fourier Series of a Dirac Train, but I'm getting after Integration by Parts that Dn equals to 0 and not to 1 as needed to be. Could you please help me find my ...
0
votes
2answers
61 views

Integration on f(-x)

Suppose that for some function $f(x)$ such that $\int_{-\infty}^{\infty} f(x)$ exists, then will that necessarily imply that $$\int_{-\infty}^{\infty} f(x)=\int_{-\infty}^{\infty} f(-x)?$$
0
votes
1answer
27 views

Homogeneous first order ODE with partial fractions solution help

I am not sure where I went wrong. I am very confident my algebra is fine and I set up my partial fractions exactly as I was supposed to. I am not sure what is wrong. $(x^2-y^2)\frac{dy}{dx}=2xy$ of ...
0
votes
0answers
31 views

Evaluate $\int{ \boldsymbol{x}^T \boldsymbol{x} \exp^{-\boldsymbol{x}^T \boldsymbol{x}} \, d\boldsymbol{x}}$

How can one proceed to evaluate the integral $\int_{-\infty}^{\infty}{ \boldsymbol{x}^T \boldsymbol{x} \exp^{-\boldsymbol{x}^T \boldsymbol{x}} \, d\boldsymbol{x}}$ This case of $\boldsymbol{x}$ ...
1
vote
1answer
167 views

Can knowledge of $\int f(x)^2 dx$ possibly be used in obtaining $\int f(x) dx$?

I have three problems of the form $\int_0^{\frac{1}{3}} f(x) \, dx$, where $f(x)=-2 \sqrt{9 x+1} \sqrt{21 x-4 \sqrt{3} \sqrt{x (9 x+1)}+1}$, and $f(x)=-42 x \sqrt{9 x+1} \sqrt{21 x-4 \sqrt{3} \sqrt{...
1
vote
0answers
74 views

Ahlfors' Complex AnalysisTheorem 1 of Chapter 4

Hello, I was going through Ahlfors' Complex Analysis where I found some difficulty to underestand the proof of the followning theorem. This is the way he explains it in the book: Update : $\Omega$ is ...
-2
votes
0answers
30 views

Integral of neither even nor odd function [closed]

We have that $M(x)$ is even function and $N(x)$ is odd function , $a$ a positive number and $b$ is a real number. $$\int_{0}^{a}e^{x}dx$$
0
votes
1answer
22 views

On the monotonicity of integrals

Let $f,g:\mathbb{R} \to \mathbb{R}$ be two functions such that $$f\left(x\right) \geq g\left(x\right), \text{ for every } x \in \mathbb{R}. \tag{1}$$ Then, by monotonicity of integrals, we have $$\...
0
votes
1answer
86 views

Integral $\int_{0}^{\infty}\frac{dy}{1+y^{2}}\log\left(\sqrt{1+y^{2}}+\sqrt{x+y^{2}}\right)$

I need help finding an analytical expression for the integral $$I(x)=\int_{0}^{\infty}\frac{dy}{1+y^{2}}\log\left(\sqrt{1+y^{2}}+\sqrt{x+y^{2}}\right), $$ where $0<x<1$. The expression can be ...
0
votes
2answers
37 views

Finding the area of a curve where one of the curves is a line

The following problem is from the book, Calculus and Analytical Geometer by Thomas and Finney. Problem: Find the area of the region bounded by the given curves. $$ y^2 = 4x, y = 4x - 2 $$ Answer: \...
0
votes
2answers
60 views

Solving the integral $\int\frac{1}{\sqrt{x^2+1}}\,dx$ [duplicate]

I need to solve $$\int\frac{1}{\sqrt{x^2+1}}$$ but I'm nowhere near the solution. I tried substituting $u = x^2 + 1$ such that $dx = \frac{1}{2x}$ yielding $$\int u^{-1/2} du = \frac{1}{2x}\frac{...
0
votes
0answers
23 views

Gauss Divergence theorem vs. calculating 'on hand'

I'm having problems with the following exercise. I need to calculate a flux through a part of a unit sphere: $$ S: \{ (x, y, z) \in \mathbb{R}^3 ; x^2 + y^2 + z^2 = 1 \land x, y, z \geq 0 \} \\\ \...
0
votes
2answers
52 views

What is the correct Substitution for this integral

$$\int_0^{1} ab^ac^{-a-1} dc$$ What would be the substitution so that this problem could be done by integration by parts would it be $~c = a~$? Also $~c ≥ b~$.
-4
votes
1answer
41 views

Solve the following Improper Integral [closed]

Solve the following Improper Integral$$\int_{0}^{1} \frac{dx}{\sqrt {(x(1-x))}}$$ Help me find out the solution !
4
votes
0answers
177 views

Show the integral identity $\int_0^1 \frac{\ln(1-x) (\ln(1+x))^2}{x} dx = -\frac{\pi^4}{240}$ [duplicate]

Show that \begin{eqnarray*} I_2=\int_0^1 \frac{\ln(1-x) (\ln(1+x))^2}{x} dx = -\frac{\pi^4}{240}. \end{eqnarray*} Motivation: I am actually interested in triple plums of the form \begin{eqnarray*}...
2
votes
1answer
44 views

Analysis of a Continuous Piecewise Function

I've recently come across a question that has completely stumped me, as follows: Given a function $f(x)$ such that $$ f(x) = \begin{cases} \frac{a}{x-1} [3 \sin (x-1) - 2 \tan (\ln x)], & \text{...
0
votes
1answer
43 views

Why do all integrals equal to zero when substituting $ u = x(x-a-b)$. [duplicate]

My friend showed me this substitution for an integral $$\int _a^b f(x)\, \mathrm dx:$$ Make substitution $ u = x(x-a-b)$. Now this changes the limit to $$\int _{-ab}^{-ab} \text {(something)}\,\...
2
votes
0answers
28 views

Volume integral on analytic manifolds

On smooth Riemannian manifolds, you can always find a smooth partition of unity: $$\{\varphi_\alpha\}_{\alpha\in\mathfrak{A}},\qquad \sum_{\alpha\in\mathfrak{A}}\varphi_\alpha=1$$ and only finitely ...
1
vote
0answers
33 views

How visualize a Volume

I need to calculate some volumes, but I can't because I'm not able to visualize them.I 'm not able to set the integrals. I'm going to show my problem: I have that volume $V:=\{\ \underline{x} \in \...
0
votes
0answers
35 views

Showing Convexity of a functional

I want to minimize a functional over the space of continuously differentiable functions. I have already a candidate and want to show the property of minimality by (strong) convexity of the functional. ...
-3
votes
0answers
19 views

Can someone give me some examples about “DIFFERENTIAL EQUATION WITH LINEAR COEFFICIENT X AND Y” [closed]

I cant find any other examples inthe internet and also in books. With the topic Differential equation with linear coefficients of x and y
1
vote
1answer
95 views

I need help with this integration: $\int_{0}^{\infty}x^{a}e^{x^{b}}e^{-(e^{x^{b}}-1)^c}dx$

I am trying to fined a closed form for this integration $$\int_{0}^{\infty}x^{a}e^{x^{b}}e^{-(e^{x^{b}}-1)^c}dx,$$ where $a,b,c>0$ I think the generalized integro-exponential ($E_{s}^{r}(z)=\...
1
vote
0answers
29 views

Parseval Theorem for a finite-valued function to the p-th power

According to Parseval Theorem (or Plancherel theorem), we have the following property. If $f(x)$ and $g(x)$ are two $L^2$ functions, and $P$ denotes the Plancherel transform, $$ \int_{-\infty}^{\...
-1
votes
1answer
49 views

Why the closed loop integral is zero

I have a integral : $\oint_L \frac{1}{r^2} dr $ Which I can write as : $-\oint_L d(1/r) $ r is here the distance on that plane from a point. Why is this equal to zero? Is it because the ...
0
votes
0answers
39 views

Are different equations of the constant of integration equatable?

I was thinking of some problems with nuclear physics. I need to derive a function of displacement and time for a particle colliding with is antiparticle. The two particles were at rest when they were ...
3
votes
3answers
58 views

Integrating velocity, how to get to this given equation?

Assuming every parameter is constant except for the variable $t$, how does the author get from to here ? When I integrate velocity myself I get $$z(t)=v_0t+\frac{v_e(m_0-qt)(\ln(1-\frac{qt}{m_0})-1)...
10
votes
2answers
233 views

Integral involving hypergeometric function $\int_0^1[{}_2F_1(\frac13,\frac23;1;x^3)]^2dx$

Question: How to prove $$I=\int_0^1\bigg[{}_2F_1\left(\frac13,\frac23;1;x^3\right)\bigg]^2dx=\frac{\sqrt3}{32\pi^5}\Gamma\left(\frac13\right)^9?$$ Source: An integral competition post of my country. ...
0
votes
0answers
25 views

Is it possible to show when the area of a polygon equals the area under its connected points?

First, let me preface w/ what my own understanding consists of. I've only ever taken classes up to Discrete Math & Differential Equations, and it has been a while since these topics have been ...
3
votes
1answer
191 views

How to evaluate $\int_0^1\frac{\ln x\ln^2(1-x)}{1+x}\ dx$ in an elegant way?

How to prove, in an elegant way that $$I=\int_0^1\frac{\ln x\ln^2(1-x)}{1+x}\ dx=\frac{11}{4}\zeta(4)-\frac14\ln^42-6\operatorname{Li}_4\left(\frac12\right)\ ?$$ First, let me show you how I did ...
0
votes
3answers
70 views

Calculating the area between two curves

The following problem is from the book, Calculus and Analytical Geometer by Thomas and Finney. Problem: Find the area of the region bounded by the given curves. $$ y^2 = 9x, y = \frac{3x^2}{8} $$ ...