All aspects of integration, including the definition of the integral and computing different types of integrals. For questions solely about the properties of integrals, don't use this tag alone! Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another ...

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-2
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3answers
30 views

What is the solution of $\int \frac{a}{a^2+x^2}dx$?

What is the solution of $\int \frac{a}{a^2+x^2}dx$? I have tried: $\frac{a}{a^2+x^2}=\frac{\frac{1}{a}}{1+(x/a)^2}=\frac{1}{a} \frac{1}{1+(x/a)^2}$ by multiplying both parts of the fraction by ...
0
votes
0answers
26 views

$\int \left(\int_a^x f(t)\, dt\right)\, dx$

For a continuous function $f(x)$ has anyone thought about $$\int \left(\int_a^x f(t)\, dt\right)\, dx$$ That is, what is the integral of an area function. These (area functions) are commonly brought ...
1
vote
2answers
60 views

What do we mean by derivative of a function? What does it tell? [duplicate]

Taking the derivative of any kind of function is easy but I don't know why we take the derivative? Like $f(x)=x^2$ has the derivative $2x$, so what does it mean? I don't know how to define ...
-4
votes
2answers
55 views

Could someone show me the steps in finding this integral? [on hold]

The integral is: (I have no idea how to use integration symbols, bear with me: $$ \int_{-\infty}^{2} 0.1 \ e^{-0.2 |x|} \;\mathrm{d}x$$ I need help ASAP. Please assist. Thank-you!
2
votes
3answers
33 views

How find this integral $\iint_{D}(x^2y+xy^2+2x+2y^2)dxdy$

let $$D=\{(x,y)|y\ge x^3,y\le 1,x\ge -1\}$$ Find the integral $$I=\dfrac{1}{2}\iint_{D}(x^2y+xy^2+2x+2y^2)dxdy$$ My idea: ...
3
votes
2answers
42 views

Closed form of $\int_0^1 \frac{\operatorname{Li}_2\left( \sqrt{t} \right)}{2 \, \sqrt{t} \, \sqrt{1-t}} \,dt $

I'm looking for a closed form of this integral. $$I = \int_0^1 \frac{\operatorname{Li}_2\left( \sqrt{t} \right)}{2 \, \sqrt{t} \, \sqrt{1-t}} \,dt ,$$ where $\operatorname{Li}_2$ is the dilogarithm ...
0
votes
5answers
84 views

How to prove that the function $f(x)=0.1\,e^{-0.2|x|} $ is a probability density, and then use it?

So here's the integral, I'm having a hard time solving it. I even tried integration software, but it didn't help: $$ I=\int_{-\infty}^{+\infty}f(x)\,dx,\qquad f(x)=0.1\,e^{-0.2|x|} $$ The question ...
1
vote
2answers
52 views

An Elliptic Integral - What's the Simplest Answer?

I have $$ \int_{0}^{2\pi}d\theta\left(R^{2}-\epsilon^{2}\right)\sqrt{R^{2}-\epsilon^{2}\sin^{2}\left(\theta\right)} $$ which Mathematica thinks is $$ ...
0
votes
1answer
19 views

Set up triple integral for volume (cylindrical coordinates)

I am given the following question Let $D$ be the region in $\mathbb{R}^3$ that lies within $x^2 + y^2 =4$, underneath the surface $z= 4- x^2 - y^2$ and above the surface $z=- \sqrt{9-x^2 - y^2}$ ...
2
votes
3answers
41 views

Finding the indefinite integral of a root function

I'm stuck on a particular integral problem. The problem is stated as: $$\int x \sqrt{2x+1} dx$$ My working thus far: $$\int x \sqrt{2x+1} dx = \frac{1}{2}x^2\frac{2}{3}(2x+1)^\frac{3}{2}$$ ...
1
vote
2answers
26 views

Finding the Limits of the Triple Integral (Spherical Coordinates)

Let $D$ be the region in $\mathbb{R}^3$ below $z=-\sqrt{x^2 + y^2}$ and above $z=-\sqrt{4-x^2 -y^2}$. Rewrite \begin{align*}\iiint \limits_D z^2 dV\end{align*} using Spherical Coordinates. I ...
0
votes
1answer
39 views

This is question about integration. I want you to check error.

Please tell me which part is wrong, or if there are better solution, please let me know. Thanks.
1
vote
1answer
21 views

Why $f (x):= \frac{1}{\sqrt{x}\left(1+\left|\ln x\right|\right)}$ only belongs to $L^2(0, \infty)$

This is a result given in Royden and Fitzpatrick (p. 143). Show that $$ \int_0^\infty \left[ \frac{1}{\sqrt{x}\left(1+\left|\ln x\right|\right)} \right]^p < \infty $$ if and only if $p=2$. That ...
4
votes
0answers
35 views

Closed form for integral $\int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy $

I'm looking for a closed form of this definite iterated integral. $$I = \int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy $$ From Vladimir ...
1
vote
1answer
19 views

Evaluate an integral involving tangent and secant. [on hold]

![I tried several methods that I could do. First method was I changed tan^2x = sec^2x-1, and then substitute secx to t, but it doesn't work. Second method was to use substitute tan^2x = v, secx = ...
1
vote
0answers
29 views

Integral equation solution

I have an integral equations of the form $ \int s R(s) =s f(s)-\int f(s)ds \tag 1$ Can we solve this integral equation for $f(s)$ interms of $s,R(s)$ ? Means $R(s)=\psi(s,R(s))$ (with out integral ...
-1
votes
1answer
19 views

Integration of exponential with a complex [on hold]

i want to prove the left side of the equation to the right side, can some one please help me with this
0
votes
0answers
50 views

How to find if an integral is possible to compute: Failing to solve integral for quadratic functional

I am trying to solve the below integral, and no computational method seems to be capable of solving this, nor can I do it by hand. Any ideas? $$\int_{t_0}^{t_1}[a(t)((2\dot{x^*}\dot{\eta} + ...
1
vote
0answers
21 views

Looking for advice with the following integral

I have the following integral to evaluate: $$ \frac{1}{f(t)}\int_0^t t^m (t + n)^o \sin(pt) \mathrm{d}t \quad m,n,o,p \in \mathbb{R}$$ I'm unable to proceed with this integral as it is non-trivial. ...
0
votes
1answer
20 views

Vitali Set: Inner Measure vs. Outer Measure

Context Nonlinearity in general of the Lebesgue integral for nonmeasurable functions reduces in some sense to inner and outer measure of nonmeasurable sets: ...
1
vote
1answer
55 views

Numerical value of $\int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy $

Could somebody give me a numerical value for this integral? $$I = \int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy $$
2
votes
2answers
64 views

computing integral without softwares: $\int \frac{2x+3}{x^2+\sqrt{1-x^2}}dx$

I was wondering if this integral can be solve without wolfram and others: $\int \frac{2x+3}{x^2+\sqrt{1-x^2}}dx$ Thanks.
6
votes
2answers
80 views

Computing in closed form $\sum_{n=1}^{\infty}\frac{\operatorname{Ci}\left(\frac{3}{4}\zeta(2) \space n\right)}{n^2}$

What tools would you recommend me for computing the series below? $$\sum_{n=1}^{\infty}\frac{\operatorname{\displaystyle Ci\left(\frac{3}{4}\zeta(2) \space n\right)}}{n^2}$$ I lack the starting ...
2
votes
3answers
65 views

How to integrate $\frac{y^2-x^2}{(y^2+x^2)^2}$ with respect to $y$?

In dealing with the integration, $$\int\frac{y^2-x^2}{(y^2+x^2)^2}dy$$ I have tried to transform it to polar form, which yields $$\int\frac{\sin^2\theta-\cos^2\theta}{r^2}d(r\cos\theta)$$ But, what ...
1
vote
1answer
44 views

Integration by parts, proving inductive case

${1\over2}\int_{-\pi/2}^{\pi/2}cos^{2n-1}(x) dx$ Inductive step: Show that the $integral={(2n-2)(2n-4)...\over (2n-1)(2n-3)...}$ for $n\ge2$ $T(n+1)$=... Attempted int. by parts using ...
2
votes
2answers
45 views

Where should I place the notorious '+c'?

Consider the following proof - $$I=\int \sin (\ln x)dx\\\iff I=\sin(\ln x)x-\int\frac{ \cos (\ln x) }{x}\cdot {x} dx \\\iff I=x\sin (\ln x)-\int\cos(\ln x)dx\\\iff I=x\sin(\ln x )-[x\cos(\ln ...
1
vote
3answers
38 views

finding an indefinite integral of a fraction

(a) Show that $\frac{4-3x}{(x+2)(x^2+1)}$ can be written in the form ${\frac{A}{x+2} + \frac{1-Bx}{x^2+1}}$ and find the constants $A$ and $B$. (b) Hence find ...
0
votes
2answers
47 views

Easy question on integrals

I have some problems understanding this inequality: $$\int_{x-\varepsilon x}^x \vartheta\left(t\right)dt \leq \vartheta\left(x\right)x\varepsilon$$ where $\vartheta\left(x\right)$ is the Čebyšëv (or ...
4
votes
1answer
80 views

Stuck on this intergral $\int^\frac{\pi}{3}_\frac{\pi}{4} \frac{\tan^2x}{x-\tan x} dx $ calculus I

$$\int^{\pi/3}_{\pi/4} \frac{\tan^2x}{x-\tan x} dx $$ this is that I have tried $$\int^{\pi/3}_{\pi/4} \frac{\frac{\sin^2x}{\cos^2 x}}{x-\frac{\sin x}{\cos x}} dx $$ $$\int^{\pi/3}_{\pi/4} ...
6
votes
6answers
486 views

Two methods to integrate?

Are both methods to solve this equation correct? $$\int \frac{x}{\sqrt{1 + 2x^2}} dx$$ Method One: $$u=2x^2$$ $$\frac{1}{4}\int \frac{1}{\sqrt{1^2 + \sqrt{u^2}}} du$$ ...
0
votes
1answer
22 views

Problem with simplifying before integration

Can someone explain to me how did the du = 6y^(-1/3)dy went into the last equation?
2
votes
0answers
10 views

Proof that maximal interval of existence exist and bounded

For each $\lambda\in \mathbb{R}$, let $\varphi_{\lambda}$ : $J_{\lambda}\rightarrow \mathbb{R}$ denote the solution to the following initial value problem: $$ ...
2
votes
3answers
37 views

Evaluate trig function integral

I was struggling to evaluate this integral: $$\int x\sin^2(4x)\;dx$$ Every time I try again I end up with a different answer, my most recent answer I came up with is $$-\frac1{12} x\cos^3(4x) + ...
-1
votes
1answer
11 views

Given a Riemann Integrable function f, calculate the values of A,B,C [on hold]

Given a Riemann Integrable function f, calculate the values of A,B,C Any help will be thankful. Thanks!
-5
votes
0answers
20 views

Change order between integral and differential calculation

Are those right? And I want to ask, in general case, when we can change the order of diff and integral: diff(integrate(L(x,y))) integrate(diff(L(x,y)))
2
votes
3answers
75 views

Evaluating the indefinite integral $\displaystyle \int 4x \sqrt{1 - x^4} dx$

I need help evaluating $$\int 4x \sqrt{1 - x^4} dx$$ What I have tried so far: Rewriting the integral as $$\int \frac{4x}{\sqrt{1 - x^4}} (1 - x^4) dx$$ $$\int \frac{4x}{\sqrt{1 - x^4}}dx - \int ...
0
votes
0answers
15 views

Integrability in bounded set

Let $A$ be a bounded open set in $\mathbb R^n$; let $f:\mathbb R^n \to \mathbb R$ be a bounded continuous function. Give an example where $\int_\bar A f$ exists but $\int_A f$ does not. Is about is ...
1
vote
3answers
51 views

Integration Formula

Previously, to integrate functions like $x(x^2+1)^7$ I used integration by parts. Today we were introduced to a new formula in class: $$\int f'(x)f(x)^n dx = \frac{1}{n+1} {f(x)}^{n+1} +c$$ I was ...
0
votes
0answers
9 views

Find a smooth function with prescribed moments

In several contexts I’ve encountered variants of the following problem : let $m_0,m_1,m_2$ be real numbers such that $0 < m_1 < m_0$ and $\frac{m_1^2}{m_0} <m_2 < m_0$. Then, show that ...
6
votes
5answers
122 views

How to find $\int|\cos x|\,dx$?

How do I find closed form for $\int|\cos x|\,dx$ for all real $x$? It can be expressed as incomplete elliptic integral of the second kind: $$\int|\cos x|\,dx=\int\sqrt{1-1^2\sin^2x}\,dx=E(x,1)$$ ...
1
vote
1answer
33 views

Integral involving exponents

How do we integrate $\int e^{C_1\frac{u^2+1}{u^2-1}} \ du\tag 1$ I could not find a proper substitution to convert it to a normal available form so that I can get a closed form of integration. $C_1$ ...
3
votes
1answer
95 views

Limit of the sum of $\gamma_k(x)=xf((k+1)x)-\int_{(k+1)x}^{(k+2)x}f(t)\mathrm{d}t$

Let $f$ be a continuous, decreasing function, with $\displaystyle\lim_{x\rightarrow\infty}f(x)=0$. Let $\gamma_k(x)=xf((k+1)x)-\int_{(k+1)x}^{(k+2)x}f(t)\mathrm{d}t,\displaystyle x>0$. Let ...
3
votes
0answers
45 views

Wicked domain of integration in a triple integral

I am dealing with a domain of integration of the form: $\left(\frac{x-y}{x+y}\right)^2+\left(\frac{y-z}{y+z}\right)^2+\left(\frac{x-z}{x+z}\right)^2\leq k$ The region looks like this (for $k=0.2$): ...
1
vote
2answers
21 views

Changing bounds on double integral

I have the following integral and with the following substitutions that I made: $$\int_{a}^{b}\int_{c}^{d}xy\sqrt{x^2+y^2}\,dx\,dy$$ $u=x^2, v=y^2$ $du = 2xdx, dv = 2ydy$ Which led me to ...
1
vote
1answer
36 views

Partial summation formula and integral

I have to prove that $\forall k \geq 1$ $$ \sum_{n\leq x} \frac{f(n)}{n} = \frac{1}{(k+1)!} \log^{k+1} x + O(\log^k x), $$ where $$ \sum_{n\leq x} f (n) = \frac{x}{k!} \log^k x + O(x\, \log^{k-1}x). ...
0
votes
0answers
15 views

A proof problem about intergral equation's root

Several days ago,my junior asked me the following problem: Let $$F\left( w \right) = \frac{1}{T}\int_0^T {M{x_C}\left( t \right)\cos \left( {tw} \right)dt} - \frac{{\sin \left( {{T_s}w} ...
6
votes
1answer
62 views

Closed-form of $\displaystyle\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}\Psi_3(n+1)$

Does the following series have a closed-form \begin{equation} \sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}\Psi_3(n+1) \end{equation} where $\Psi_3(x)$ is the polygamma function of order 3. Here is ...
0
votes
1answer
41 views

Integral of $\int\frac1{x}\sqrt[3]{\frac{1-x}{1+x}}dx$

I could substitute $t=\sqrt[3]{\frac{1-x}{1+x}}$ and get $\int\frac{6t^3}{t^6-1}dt$, which leads to partial fractions decomposition with 6 variables. That's annoying and may lead to mistakes. Is there ...
-1
votes
1answer
100 views

Integral of $x^x$ [duplicate]

I can't find this integral around here, does anybody suggest how to calculate this integral? $$ I = \int x^x dx. $$ Thanks in advance
0
votes
0answers
41 views

integration involving imaginary terms

How do we integrate forms of following type with imaginary terms involved? Can we get a closed form of it as result? ...