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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31 views

Asymptotic approximation inside of an integral

Suppose we want to find $$ \lim_{n \rightarrow \infty} \int_{a}^b f_n(x) dx $$ for $a, b \in \mathbb{R}$, $a \le b$. Suppose there is a sequence of functions $g_n(x)$ such that $g_n(x) \sim f_n(x)$ ...
3
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2answers
160 views

How to prove Raabe's Formula [duplicate]

For quite some time, I've been trying to prove Raabe's Formula, or in other words: $$\int_a^{a+1} \ln\bigg(\Gamma(t)\bigg)dt=\dfrac{1}{2}\ln(2\pi)+a\ln(a)-a$$ This is how I tried: ...
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1answer
121 views

How to integrate $\int^{\infty}_{-\infty} e^{-2\pi^2/x^2} dx$?

I am wondering how can i integrate this quantity above? Here it is again, $$\int^{\infty}_{-\infty} e^{-2\pi^2/x^2}dx.$$ Thanks a lot.
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1answer
34 views

How do I calculate values for Gamma function with complex arguments?

I can calculate the values of Gamma function for positive integer arguments using the formula $\Gamma (t) = \displaystyle\int_0^{\infty} e^{-x} x ^ {t-1} \mathrm{d}x $. Which is equal to $ (t-1)! $. ...
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1answer
29 views

Calculating the volume of a solid of revolution about a line.

A figure is formed by revolving the region bounded by $f(x) = \cos{(x)}$ and $g(x) = \sin{(x)}$ from $0$ to $\dfrac{\pi}{4}$ about the line $y=-1$. This figure is formed by integration of two ...
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1answer
58 views

Integral with complex variable

I want to compute $$ \int_{-\infty}^{\infty} \frac{1}{\sqrt{x+yi +2}} dy $$ where $i$ is the imaginary number. How to compute this integral??
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4answers
155 views

To check the convergence of an integral (2)

I tried to check if this integral is convergent: $\int _{-\infty }^{\infty }\left(\frac{\sin\left(x\right)\ln\left|x\right|}{1+x^2}\right)dx\:$ so, there are 3 points to check: $\pm\infty$ and $0$. ...
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1answer
66 views

Is there a short symbol that denotes integration?

I want to illustrate partial integration, see below. With derivatives we can just write $(term)'$. Is there something similar for integration? The best I could come up with is $\int(term)\mathrm{d}x$. ...
2
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2answers
61 views

To check the convergence of an integral

I tried to find out if this integral is convergent or divergent, $$\int _0^{\frac{\pi}{2}}\left(\frac{\ln\left(\sin x\right)}{\sqrt{x}\:}\right)\:dx$$ I know that the problematic point is near ...
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0answers
52 views

Computing $\int_{0}^{1}\int_{0}^{1}\int_{\max\{x,y\}}^{1} e^{z^3} dz dx dy$

I am trying to compute $$\int_{0}^{1}\int_{0}^{1}\int_{\max\{x,y\}}^{1} e^{z^3} dz dx dy$$ What I have done is to reverse the order of integration; so I did the integration with respect to $x$ and ...
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3answers
54 views

what is the value of $\int \sin(x)\cos(x)dx$? $\frac{\sin^2(x)}{2}$ or $\frac{-\cos^2(x)}{2}$ or $\frac{-\cos(2x)}{4}$

$\int \sin(x)\cos(x)dx = \frac{\sin^2(x)}{2}$ because $$\frac{d}{dx}\frac{\sin^2(x)}{2}=\sin(x)\frac{\sin(x)}{dx}=\sin(x)\cos(x)$$ but also ...
2
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2answers
48 views

Help with indefinite integration

I am learning indefinite integration, yet am having problems understanding and recognizing where to substitute what. a good trick is to attempt convert algebraic expressions into trigonometric and ...
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1answer
24 views

Finding the volume of a cone with and oblique base.

The base of $S$ is an elliptical region with boundary curve $9x^2+4y^2=36$. Cross-sections perpendicular to the $x$-axis are isosceles right triangles with hypotenuse in the base. The base of $S$ is ...
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0answers
60 views

Multiple integrals of dirac delta

I'm working on a problem where integrals of this form arise: $$ \int\limits_{x_1=-1}^1 \;\; \int\limits_{x_2 =-1/3} ^{1/3} \dots \int\limits_{x_n =-1/(2n+1)} ^{1/(2n+1)} ...
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1answer
69 views

If $k$ is a non-zero constant, determine by inspection the indefinite integral of $\int e^{kx} dx$.

I have to solve this exercise: If $k$ is a non-zero constant, determine by inspection the indefinite integral of $\int e^{kx} dx$. By inspection, I guess it means that it should be solved by ...
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2answers
59 views

Problem with understanding a Differential in Multivariable Calculus

I have just started with Partial Differentiation and the book from where I'm learning (Mathematical Methods in the Physical Sciences) had the following problem on approximations using differentials ...
2
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1answer
85 views

Hint to integrate $\int_0^1 \frac{\log(1+x)}{x} \, \mathrm{d}x$?

How to integrate $$\int_0^1 \frac{\log(1+x)}{x} \, \mathrm{d}x\text{ ?}$$ The answer is $\frac{\pi^2}{12}$, but I don't seem to get a way to reach there.
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1answer
21 views

setup of volume using double integrals

So i have to setup a double integral for the volume of a solid under $z = 3x^2 +y^2$ and over the region bounded by $y = -x$ and $y = x^2 -6$ I tried equating y so that left me with $x^2 +x -6=0$ ...
5
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1answer
65 views

double integral problem $\iint e^{\frac{x}{x+y}}dxdy$

I'm trying to integrate $$\iint e^{\frac{x}{x+y}}dxdy$$ where $y \leq (1-x)$ and $0 \leq x,y \leq 1$. I tried to define new variables as $u=x$ and $v=x+y$, but I can't solve this either. I have ...
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2answers
55 views

Integral of logarithm of exponential function

I am trying to solve this integral: $$\int \log\left(1 + \frac{1}{\pi}\exp\left(\frac{-x^2}{2a^2}\right)\right) dx$$ where $a$ is some fixed constant. The bounds of this integral are $-a$ and $a$, ...
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1answer
57 views

How can I solve the integral below using complex variables?

How can I solve the integral below using complex variables? $$ \int\limits_{0}^{2\pi}\sin\frac{\theta}{2}\;\mathrm{d}\theta $$ I know how to solve the integral of $\sin θ$. I replace it by ...
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1answer
71 views

Multiple Integrals

$$\int _{ 5 }^{ 20 }{ \int _{ 5 }^{ 20 }{ \int _{ 5 }^{ 20 }{ \int _{ 5 }^{ 20 }{ \ln(w+x+y+z) }\ dw\; dx\; dy\; dz } } }$$ Unfortunately I cannot think of how to approach this problem. The only ...
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2answers
84 views

Integral of $\int \frac{x+1}{(x^2+x+1)^2}dx$

I'm currently learning Calculus II and I have the following integral: Integal of $\large{\int \frac{x+1}{(x^2+x+1)^2}dx}$ I've tried with partial fractions but it led nowhere, I've tried with ...
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1answer
61 views

Calculating the volume of cutted cone by a plane

I need to calculate the volume bounded by the plane: $x+y+z=5$ and by the cone $z^2 = x^2 + y^2$, som my V that i'm $dv$-ing on it is cutted cone in non simetric way (i can find the equation of the ...
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1answer
59 views

Sum on integration and binomial theorem.

If $(1+x)^n = \sum_{r=0}^n \binom{n}{r}x^r$ and $$\sum_{r=0}^n \frac{(-1)^r}{(r+1)^2} \binom{n}{r} = k\sum_{r=0}^n \frac{1}{r+1}$$ Then prove that $$k=\frac{1}{n+1}.$$
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0answers
27 views

Integrating over particular grids to obtain Spherical Harmonic coefficients

Theoretically the spherical harmonic expansion coefficients of a function $f$ should be calculated via a continuous integration: $$F_{lm} = \int_{0}^{2\pi}\int_{0}^{\pi} ...
2
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1answer
38 views

Combining Fubini and Tonelli's in one single Assumption

I am referring to the statements on Wikipedia, there it is said that Fubini's Theorem states that if $f : X\times Y \to \mathbb R$ is integrable, then $$ \int_X \left( \int_Y f(x,y) dy\right) dx = ...
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67 views

If $\lim_{x\to\infty}\left(f(x)+\int_{0}^xf(t)dt\right)$ exists, what about $\lim_{x\to\infty}f(x)$? [duplicate]

Given that $f(x)$ is continuous on $[0,\infty]$. If $\lim\limits_{x\to\infty}\left(f(x)+\int_{0}^xf(t)dt\right)$ exists then evaluate $\lim\limits_{x\to\infty}f(x)$
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2answers
75 views

Integral of monomial and logarithm: is this true? $\lim_{k\to -1}\frac{x^{k+1}}{k+1} = \log|x|$

It is well know that: $$\int x^k \text{d}x = \begin{cases} \displaystyle\frac{x^{k+1}}{k+1} + c & k \neq -1\\ \\ \log|x| + c & k = -1\end{cases}$$ My guess is: $$\lim_{k\to ...
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1answer
27 views

Partial Integration for measures

I have the following formula in mind, $\mu$ a measure on $\mathbb{R}$. Any sigma-finite measure on $\mathbb{R}$ can be decomposed into a absolut continuous part, a "point measure" and a singular ...
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2answers
197 views

How to compute the integral $ I\left(c\right)=\int_{0}^{1}{\frac{\ln(1-cx)}{1+x}dx} $

I am currently working on this question and the following integral came up: $$ I\left(c\right)=\int_{0}^{1}{\frac{\ln(1-cx)}{1+x}dx} $$ for a suitable c. I would like to compute it in terms of ...
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3answers
49 views

How derivatives of integrals like $G(x)=\int_{x-\sin x}^{\sin x}\arcsin(t)dt$ are computed?

I need to find the derivative of $G(x)=\int_{x-\sin x}^{\sin x}\arcsin(t)dt$. I know I don't have to find the integral, but I just have troubles computing the derivative. I suppose I get: ...
3
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1answer
103 views

Evaluate $\int_0^a \frac{1}{x}\mathrm dx$

This would be $\ln(a)-\ln(0)$. Because $\ln(0)$ is undefined, is this integral undefined too?
3
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1answer
53 views

Evaluate $\int \dfrac{2\pi y}{2y^3-1}dy$

Evaluate: $$\int \dfrac{2\pi y}{2y^3-1}dy$$ I've been struggling with this for a while. If it had just been $y^3$ instead of $2y^3$ in the Denominator, Partial Fraction Decomposition, although ...
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2answers
53 views

Definite integral of product of exponential function and trigonometry function.

Let $x_0$ and $\sigma$ be constants. How do we evaluate the following? $$ \large \int^{L}_{-L}e^{-\frac{(x-x_0)^2}{2\sigma^2}}\cos x \, \mathrm{d}x $$ I think I can solve that with integration by ...
5
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2answers
210 views

Prove that $\lim_{t\to \infty} t\mu(\{x:f(x)\geq t\})=0$

Problem Suppose $f$ is a non-negative integrable function on a measure space $(X,\mathcal{A},\mu).$ Prove that $$\lim_{t\to \infty} t\mu(\{x:f(x)\geq t\})=0$$ Attempt Let $E_t=\{x:f(x)\geq t\}$ ...
10
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4answers
633 views

Integral becomes improper after a substitution

I'm suprised about the following phenomenon which I would like to discuss with you. Consider the proper integral $$\int_{\pi/4}^{\pi/2}\frac{1}{\sin(x)}dx.$$ Since $\sin(x)$ is a diffeomorphism on ...
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1answer
65 views

Difficult Integration Question (Reverse Chain Rule)

My friend gave me this integration question, after after a while on it, I am not sure how to go about solving it: $$\int\frac{dx}{x(\ln{x} + 1)^2}$$ Any suggests are welcome, I'm really lost at this ...
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5answers
396 views

Definite integral of even powers of Cosine.

I'm looking for a step-by-step solution to the following integral, in terms of n$$\int_0^{\frac{\pi}{2}} \cos^{2n}(x) \ {dx}$$I actually KNOW that the solution is$${\frac{\pi}{2}} \prod_{k=1}^n ...
4
votes
3answers
144 views

Finding $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{1}{1+\sqrt{\tan x}}dx$ [duplicate]

How can I integrate $$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{1}{1+\sqrt{\tan x}}dx\ \ \ ?$$ I have made the integral into the form of $\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}$, but ...
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2answers
47 views

integrable function in Lebesgue sense

How we prove that $\displaystyle{f : x \mapsto \frac{x}{e^{x}-1}}$ is Lebesgue integrable in $\mathbb{R}_{+}$?
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5answers
2k views

Why do un-integrable funcitons exist?

By un-integrable I mean functions whose antiderivative can not be expressed in terms of elementary functions. I recently learnt that any differentiable function can be expanded using the Taylor ...
3
votes
1answer
123 views

Evaluate $\int_0^1 \frac{\ln \left(a+\sqrt{a^2+1}\right)}{a\sqrt{a^2+1}}da$

Problem: Evaluate$$\int_0^1 \dfrac{\ln \left(a+\sqrt{a^2+1}\right)}{a\sqrt{a^2+1}}da$$ Unfortunately I have absolutely no idea as to how to approach this problem. It was suggested that I try ...
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0answers
36 views

Evaluate: $\int_0^1 \frac{1-x}{(1+x)\log(x)}\, dx$ [duplicate]

Evaluate: $$\int_0^1 \frac{1-x}{(1+x)\log(x)}\, dx$$ Wolfram alpha gives an answer of $\log(2/\pi)$, but I have been unable to prove this. Help appreciated.
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4answers
60 views

How to proceed with evaluating $\int\frac{dx}{\sqrt{9+4x^2}}$ and $\int\tan^2(3x)dx$

$\displaystyle\int\frac{dx}{\sqrt{9+4x^2}}$ $\displaystyle\int\tan^2(3x)dx$ For the first one i'm not sure if I did it correctly, here is what I did: Let $2x=3\tan(t)$, so $x=\frac{3}{2}\tan(t)$ ...
2
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2answers
144 views

Solve this Integral for $a$ where $a$ is a real number

$$\large{\int_0^1 ((1-x^a)^\frac{1}{a}-x)^2 dx}$$ can anybody solve this step by step please?
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1answer
23 views

Clarification about Asymptotic comparison test for Improper integrals

If I have an improper integral $\displaystyle \int_{a}^{b}f(x)dx$, and $b$ is the improper extrem, if $f(x)\sim g(x)$ for $x->b^-$, the integrals $\displaystyle \int_{a}^{b}f(x)dx$ and ...
2
votes
2answers
79 views

Help with the integral $\int_{0}^{\infty}\frac{y^{2}e^{y}}{e^{sy}+e^{-sy}-2}dy$

I want to do the integral : $$I(s)=\int_{0}^{\infty}\frac{y^{2}e^{y}}{e^{sy}+e^{-sy}-2} \, \mathrm{d}y$$ $s$ being a complex parameter. I tried expanding the dominator of the integrand, but this way ...
1
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1answer
125 views

Change of variable formula for the image of a hypercube

Let $\varphi: \mathbb{R}^n\to \mathbb{R}^n$ be an injective $C^1$ map. Let $I=[0, 1]^n$. I want to show that $$m(\varphi(I))=\int_I \left|\det D\varphi(x)\right|dx.$$ This is a special case of the ...
2
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0answers
38 views

Smart coordinates for six-dimensional integral

I have a (hopefully) simple question: I am dealing with a definite (on all of $\mathbb{R}^6$) six-dimensional integral $$\int_{\mathbb{R}^6} F(\vec{x}_1,\vec{x}_2)d^3x_1d^3x_2$$ where the function ...