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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Integral formulation for LDE

I am trying to put the system in a integral formulation. All goes well for the first integration as I obtain What I don't know is how to perform the second integration in this last term. My ...
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2answers
51 views

Indefenite Integral requiring substitution

Can someone please help me find a useful substitution for the following integral: $$\int \frac{1}{\sqrt{x}(1+\sqrt{x})^2}dx$$ I tried letting $ u = \sqrt{x} $ But I couldn't proceed. Please help.
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2answers
57 views

Can I integrate $\frac{x}{1-x}$ by substitution?

I saw a person use substitution like this: $$\int \frac{x}{1-x} dx$$ Let $u= (1-x)$, $x= 1-u, du= -1\cdot dx$ $\Rightarrow$ $-du=dx$ $$\int \frac{1-u}u (-du)$$ Can I use substitution like this? I ...
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1answer
40 views

Another parametric integral relating to hyperbolic function

if $0<a\leq1$, then canwe get a closed form of $$I(a)=\int_0^\infty\frac{x}{\tanh x}\frac{1}{\cosh^2(ax)}dx.$$ In fact,if $a=1$,$I(a=1)=\pi^2/8$.
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2answers
45 views

parametric integral relating to hyperbolic function

Suppose that $a$ is real number such that $0<a<1$, how can we calculate $$ I(a)=\int_0^\infty \big(1-\frac{\tanh ax}{\tanh x}\big)dx .$$ As for some speical cases, I can work out $I(1/2)=1$. ...
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2answers
40 views

How to find $P(X>x)$ when the density is known but the integral does not seem to converge

I am trying to evaluate $$P(X>x) = \int_x^{\infty } t^{\kappa } \exp{\left(-\rho t^{\alpha\kappa + 1}\right)} \, dt$$ where $\kappa$, $\rho$ and $\alpha$ are all constants. I have tried some ...
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3answers
36 views

This is the question about integration. I want to know how to approach this question. [duplicate]

My solution makes same loop, which eventually makes the equation as 0 = 0 form.
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1answer
32 views

Integration by Parts and Convergent/Divergent Series Calculus

We are asked to integrate: $$\int x (lnx) dx$$ Integration by parts gives us: (using L-I-A-T-E) $$u = lnx$$ $$ du = (1/x)dx$$ $$ dv = xdx $$ We find v by integrating dv: $$ v = (1/2)x^2 $$ ...
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1answer
27 views

Integral of [(1+2y^2)/(3-y)]dy (obtained from a differential equation)

This question actually arises from this Differential Equations question: Find the family of solutions for: (1+2y^2)(dy/dx) + (3-y)cosx = 0 I ruled out the methods I've so far learned in class ...
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2answers
42 views

Integration of $1/(x^2+x\sqrt{x})$

The question is to evaluate $\displaystyle7\int\frac{dx}{x^2+x\sqrt{x}}$. My solution is attached. The problem of my solution is if I use partial fraction, loop will be made, and this makes ...
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2answers
42 views

Integration of $(5x^2+2x-5)/(x^3-x)$

The problem is to evaluate $\int \frac{5x^2+2x-5}{x^3-x}\,dx$. This is the solution that I tried: I really have no idea of this problem. After check my solution, if there are any problem that ...
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2answers
51 views

This is the question about integration.

My idea is to use substitute integration. Since there is square root of (1-x^2), I made x = cos^2t, and then eliminated square root. I don't know why my answer is wrong. I already conducted ...
4
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0answers
57 views

Closed form for $\int_1^\infty\frac{dx}{\Gamma(x)}$

Is a closed form for $$\int\limits_1^{+\infty}\frac{dx}{\Gamma(x)}$$known? I tried to find it, but all well-known integrals involving gamma-function (such as of $\log\Gamma(x)$ or the like) don't ...
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6answers
66 views

For polynomials $f,g$, why is $\int_0^\infty \frac{fg}{e^x}\, dx$ absolutely convergent?

Why does the integral $\displaystyle \int_0^\infty \frac{fg}{e^x}\, dx$ have to be convergent for all real polynomials $f$ and $g$? Can anybody give me a proof?
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3answers
42 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 ...
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0answers
33 views

$\int \left(\int_a^x f(t)\, dt\right)\, dx$ [on hold]

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
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2answers
74 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 ...
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2answers
63 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!
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3answers
39 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
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2answers
68 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 ...
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5answers
86 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 ...
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2answers
56 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 $$ ...
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1answer
21 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
47 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: $$\int x \sqrt{2x+1} dx = \frac{1}{2}x^2\frac{2}{3}(2x+1)^\frac{3}{2}$$ ...
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2answers
27 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 ...
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1answer
45 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.
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1answer
26 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 ...
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0answers
50 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 ...
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1answer
23 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 = ...
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0answers
30 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 ...
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1answer
21 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
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0answers
52 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} + ...
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0answers
22 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. ...
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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: ...
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1answer
58 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? $$I = \int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy $$
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2answers
66 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.
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2answers
104 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
67 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 ...
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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
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2answers
47 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 ...
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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 ...
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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 ...
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1answer
82 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
489 views

Two methods to integrate?

Are both methods to solve this equation correct? $$\int \frac{x}{\sqrt{1 + 2x^2}} dx$$ Method One: $$u=2x^2$$ $$\frac{1}{4}\int \frac{1}{\sqrt{1^2 + \sqrt{u^2}}} du$$ ...
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1answer
23 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
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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
38 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) + ...
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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!
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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 ...