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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1answer
281 views

evaluate $\int \frac{x^2\cdot\log(x)}{x+1} dx$

I am trying to evaluate integral: $$\int \frac{x^2\log(x)}{x+1}dx$$ But I have some problems with it. If I use Wolfram Alpha like this I get a result, but I need evaluate it by hand. Which ...
0
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2answers
58 views

Equivalence of multiple integral

How do I prove that for any function $f$, any $n$, and any $c$, $$ \frac{1}{n} \cdot \frac{1}{c^{n} } \int _{0}^{c}\int _{0}^{c}\cdot \cdot \cdot \int _{0}^{c}f(x_{1} )+f(x_{2} )+...+f(x_{n} )dx_{n}...
4
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1answer
271 views

Is $f$ non-decreasing a.e. if its primitive is convex?

The subsequent statement can be regarded as a follow-up to If $\int_0^x f \ dm$ is zero everywhere then $f$ is zero almost everywhere Is $f$ non-negative a.e. if its primitive is non-decreasing? ...
2
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2answers
283 views

What could be the answer for this $\int \limits_{-1}^{1}\big(x\sin{\pi}x\big)dx$??

$\displaystyle\int \limits_{-1}^{1}\big(x\sin{\pi}x\big)dx $? a) 2 b) -2 c)1 d)0 I've reduced it to $=2\displaystyle\int \limits_{0}^{1}\big(x\sin{\pi}x\big)dx$ as it is an even function. Do I ...
11
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2answers
867 views

How to evaluate this integral?

How would I prove that$$\int_{0}^\infty \frac{\cos (3x)}{x^2+4}dx= \frac{\pi}{4e^6}$$ I changed it to $$\int_{0}^\infty \frac{\cos (3z)}{(z+2i)(z-2i)}dz$$, and so the two singularities are $2i$ and $-...
29
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2answers
1k views

Proving $ \int_{0}^{\infty} \frac{\ln(t)}{\sqrt{t}}e^{-t} \mathrm dt=-\sqrt{\pi}(\gamma+\ln{4})$

I would like to prove that: $$ \int_{0}^{\infty} \frac{\ln(t)}{\sqrt{t}}e^{-t} \mathrm dt=-\sqrt{\pi}(\gamma+\ln{4})$$ I tried to use the integral $$\int_{0}^{n} \frac{\ln(t)}{\sqrt{t}}\left(1-\frac{...
24
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3answers
1k views

Showing that $ \int_{0}^{1} \frac{x-1}{\ln(x)} \mathrm dx=\ln2 $

I would like to show that $$ \int_{0}^{1} \frac{x-1}{\ln(x)} \mathrm dx=\ln2 $$ What annoys me is that $ x-1 $ is the numerator so the geometric power series is useless. Any idea?
3
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2answers
443 views

Do there exist intervals in $[0,1]$ of rational numbers?

Basically I'm struggling to prove that if you take the indicator function $f$ on $\mathbb{Q}$ intersect $[0,1]$ and $\phi, \psi$ are step functions such that $\phi \leq f \leq \psi$ then $\phi \leq 0$ ...
1
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2answers
89 views

Integral Question (with a possible error)

I've been assigned the following question: Prove that if $g$ is integrable on $[a,b]$, then $\int_{a}^{b}g(x)dx=\int_{a}^{b}g(a+b-x)dx.$ I am stumped because it seems to me that $$\int_{a}^{b}g(...
4
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2answers
214 views

Is there a formula for solving integrals of this form?

I was wondering if there was a substitution formula to solve integrals of this form: $\int f(g(x))g''(x)dx$
17
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2answers
4k views

$\int_0^{\infty } \frac{\log (x)}{e^x+1} \, dx = -\frac{1}{2} \log ^2(2)$ How to show?

$$ \int_0^{\infty } \frac{\log (x)}{e^x+1} \, dx = -\frac{1}{2} \log ^2(2) $$ Anyone an idea on how to prove this?
3
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4answers
690 views

Calculating this residue (residue theorem)

Given $$\int_{\gamma}\frac{1}{(z-a)(z-\frac{1}{a})}dz,$$ and $0<a<1$, where $\gamma(t)=e^{it}$ and $0\le t \le 2\pi$ I am trying to find the residue of$$f(z)=\frac{1}{(z-a)(z-\frac{1}{a})}$$ ...
3
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1answer
398 views

Evaluating this contour integral

I am trying to show that $$\frac{1}{2\pi i}\int_{\gamma(0,3)}\frac{e^{(zk)}}{z^2+1}\mathrm dz = \sin k$$ for all $k \in \mathbb{C}$. And $\gamma(z_0,R)$ is the circular contour $z_0+Re^{it}$, where $...
33
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8answers
11k views

Which is the easiest way to evaluate $\int \limits_{0}^{\pi/2} (\sqrt{\tan x} +\sqrt{\cot x})$?

Which is the easiest way to evaluate $\int \limits_{0}^{\pi/2} (\sqrt{\tan x} +\sqrt{\cot x})$? I have reduced this problem to $$ 2\int_0^{\pi/2} \sqrt{\tan x} \ dx$$ but now, evaluating this ...
2
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1answer
2k views

Integrable Odd Function

I am working on the following question: Let $f$ be an integrable odd function, i.e. $f(-x)=-f(x)$ for all $x\in \mathbb{R}$. Show algebraically that $\int_{-a}^{a}f(x)dx=0$ for all real numbers $a$....
2
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1answer
207 views

real-value solution

I have this integral $$\int\frac{dz}{\sqrt{(z^{2}-\rho^{2})(\lambda^{2} - z^{2})}}$$ and parameters obey the following conditions $$z= \exp[k\varphi],$$ $$\lambda^{2} = \frac{b + \sqrt{b^{2} - 4ac}}{...
1
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2answers
104 views

Evaluating $\int y'(x) \cdot (y(x) +1)^2 \cdot dx$

I evaluate the following integral with pen and paper: $$\int y'(x) \cdot (y(x) +1)^2 \cdot dx$$ And I get to the following result: $$\frac{(y(x) + 1)^3}{3}+C$$ However, after that I went to ...
10
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2answers
416 views

integrating the secant function, who figured this out?

I was looking at how the secant function is integrated. The process is not obvious, and I don't expect it to be but I wanted to know if anyone knows who figured this out. Here's what I'm talking ...
2
votes
1answer
415 views

integral resulting in Bessel function

Prove that $$\int_{0}^{\infty} \sin \left(x\right) \sin \left(\frac{a}{x}\right) \ dx = \frac{\pi \sqrt{a}}{2} J_{1} \left( 2 \sqrt{a} \right)$$ where $J_{1}$ is the Bessel function of the first kind ...
9
votes
1answer
3k views

Why is the Riemann integral only defined on compact sets?

Every text I look at says a function must be bounded and be defined on a compact set before one can even think about the Riemann integral. Boundedness makes sense, otherwise the Darboux sums could be ...
8
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1answer
368 views

Limit of $\frac1{c^n}\iint_{[0,c]^n}\frac{f(x_1) +f(x_2) +\cdots+f(x_n)}{g(x_1) +g(x_2) +\cdots+g(x_n)}\,dx_1dx_2\cdots dx_n$ when $n\to\infty$

A. How do I prove the following sequence converges as $n$ goes to $\infty$ for any $c$, and how do I find the limit? $$ \begin{align} a_1 &=\frac{1}{c} \int _0^c\frac{x_1 }{1+x_1}\;dx_1 \\ \\ ...
0
votes
1answer
547 views

Manipulating a product term inside an integral

I have an expression of the form $$P(x) = \int_0^\infty \prod_{i=0}^{n} e^{-G(a, x_i)}\,\mathrm{d}a $$ and I was wondering if there was any way that I could swap the order of the product and the ...
3
votes
1answer
112 views

What was done in this equation involving the fundamental theorem of calculus?

All I know is that it uses the fundamental theorem of calculus. $$\large\frac{d}{dx}\int_{x^2}^{\sin x} e^{xt^2}dt = e^{x\;\sin^2 x}\cos x - e^{x^5}2x+\int_{x^2}^{\sin x} t^2e^{xt^2}dt$$
6
votes
1answer
221 views

how to evaluate a definite integral (looks almost like nonintegral moments of a Gaussian)

I'd like to show the following equality (at least Mathematica claims it is an equality): \begin{multline*} \int_0^\infty x^p \exp(-(ax - b)^2)\, dx = \frac{1}{2} e^{-b^2} a^{-p-1} \left(\Gamma \left(\...
2
votes
1answer
2k views

Integral of $x^\alpha$ for Various $\alpha$

How can I prove this? I know why it's the case, but I can't prove it. Give all $\alpha \in \mathbb R$ such that: a) $\displaystyle\int_a^b x^\alpha dx$ is convergent b) $\displaystyle\int_1^\infty ...
18
votes
5answers
2k views

How to prove that $\lim \frac{1}{n} \sqrt[n]{(n+1)(n+2)… 2n} = \frac{4}{e}$

I'd like a hint to show that: $$\lim \frac{1}{n} \sqrt[n]{(n+1)(n+2) \cdots 2n} = \frac{4}{e} .$$ Thanks.
3
votes
1answer
183 views

General formula of $I_{2n} = \int_{-\infty}^{+\infty} e^{-x^{2n}}dx$

$I_{2n} = \int_{-\infty}^{+\infty} e^{-x^{2n}}dx$ We know very well $I_{2}= \sqrt{\pi}$ Could you please help me find general formula of $I_{2n}$? Thanks for answers
5
votes
3answers
270 views

Approximate $\int_{0}^{\infty} \frac{\text{d} x}{1 + x^4}$

Now, I have been given this integral. And need to approximate it. My first idea was to use a Taylor series, but this series explodes, as x reaches infinity. Does anyone know how to approximate ...
1
vote
1answer
85 views

representing arcsinh as a logarithm

I am trying to understand this equality: $$ \ln{\left|\frac{x}{2}+\sqrt{\frac{x^2}{4}+1}\right|} + C= \ln{|x+\sqrt{x^2+4}|} + C'$$ My teacher didn't really explain it, she just noted that "the ...
2
votes
2answers
3k views

Integrate $\log(x)$ with Riemann sum

In a homework problem I am asked to calculate $\int_1^a \log(x) \mathrm dx$ using a Riemann sum. It also says to use $x_k := a^{k/n}$ as steps for the stair functions. So far I have this: My step ...
2
votes
2answers
146 views

Series expansion of $ \int_{0}^{1} \frac{x^n}{1+x^n} \mathrm dx $

I have proved $$ \int_{0}^{1} \frac{x^n}{1+x^n} \mathrm dx \sim \frac{\ln(2)}{n}$$ How can I get further and find $ a$ such that: $$ \int_{0}^{1} \frac{x^n}{1+x^n} \mathrm dx=\frac{\ln(2)}{n}+\frac{...
2
votes
1answer
142 views

Approximation of $ \int_{0}^{\infty} \frac{\ln(t)}{\sqrt{t}}e^{-nt} \mathrm dt,n\rightarrow\infty $

How can I find the first term of the series expansion of $$ \int_{0}^{\infty} \frac{\ln(t)}{\sqrt{t}}e^{-nt} \mathrm dt,n\rightarrow\infty ?$$ Or: As $$ \int_{0}^{\infty} \frac{\ln(t)}{\sqrt{t}...
2
votes
2answers
270 views

Is $f$ non-negative a.e. if its primitive is non-decreasing?

Let $f:[a,b]\to\mathbb{R}$ be Lebesgue integrable. Clearly, if $f$ is non-negative then $$ g:[a,b]\ni x\mapsto\int_a^x f(t)\,\mathrm{d}t\in\mathbb{R} $$ is non-decreasing since for $x<y$ it ...
2
votes
2answers
2k views

Evaluating $\int\sin^3t \, dt$

I have this integral: $$\int\sin^3t \, dt$$ I have tried partial integration with $\sin t \cdot \sin^2t$, but then I get another integral to evaluate which needs partial integration: $$\dots \int \...
3
votes
4answers
184 views

simple real analysis question on integration

it is trivial that $\int_0^{2\pi} \cos(x)\,dx = 0$. Intuitively, it is clear that for a strictly decreasing positive function $f(x)$, $$ \int_0^{2\pi} f(x) \cos(x)\,dx \ge 0 $$ but I have no glue ...
-2
votes
1answer
150 views

how to code PDF ? or for that matter Integration in general.

I need to use the Probability Function on a set of values. I am okay with understanding the math, but i don't quite know how to convert that stuff to code. basically i need the integral of ...
2
votes
1answer
708 views

Fourier transform of function involving $\log$

I found the following problem which I am unable to solve. Calculate the following integral $$\int_{\mathbb{R}} \frac{d\omega}{2\pi} \log (1 + i a/\omega ) e^{-i \omega t}$$ for $a>0$ and $t\...
0
votes
1answer
191 views

Integral Equation & Picard's Theorem further work

Following on from my question here about the integral equation $$y(x)=1+\int^{x}_{0}(\tanh s)y(s)ds$$ we now look to appeal to Picard's Theorem. Let $\{y_n\}_{n \geq 0}$ be the sequence of Picard ...
1
vote
0answers
85 views

Means of sequence of functions

Let $(\Omega_n)$ be a sequence of subsets of $\mathbb{R}^d$ with $\Omega_n\uparrow\mathbb{R}^d$ where the Lebesgue measure $\lambda^d(\Omega_n)$ is finite for every $n$. Let $(f_n)$ be a sequence of ...
27
votes
5answers
31k views

Integral of periodic function over the length of the period is the same everywhere

I am stuck on a question that involves the intergral of a periodic function. The question is phrased as follows: Definition. A function is periodic with period $a$ if $f(x)=f(x+a)$ for all $x$. ...
4
votes
1answer
1k views

Integral $\int_{0}^{1}(\sqrt{2-x^2}-\sqrt{2x-x^2})dx$

I am working on the following integral problem: $$\int_{0}^{1}(\sqrt{2-x^2}-\sqrt{2x-x^2})dx$$ There is a hint as well, which suggests interpreting the definite integral as the area bounded by ...
3
votes
1answer
283 views

How to compute the integral $\frac{1}{2\pi i} \int^\infty_{-\infty} \frac{e^{ixy}}{y - i} \, dy$

I am reading an essay that says that it is true that \begin{equation} \frac{1}{2\pi i} \int^\infty_{-\infty} \frac{e^{ixy}}{y - i} \, dy = \begin{cases} e^{-x} & \text{for x }>0 \\ ...
2
votes
1answer
173 views

How do you solve this simple integral?

What is $\int \frac{4t^2}{5t-5}dt$? I understand I can take out the constant and divide the polynomials to simplify the function. But if I didn't do the division, how then would I integrate this?
1
vote
2answers
1k views

Correct integration formula

I'm the author of application for android called Calculator++ One of the users submitted a bug in the integration of trigonometric functions. And I'm absolutely disagree with him. In my app I use ...
13
votes
3answers
1k views

What contour should be used to evaluate $\int_0^\infty \frac{\sqrt{t}}{1+t^2} dt$

Could anyone help me decide what contour to use to evaluate this integral? $$\int_0^\infty \frac{\sqrt{t}}{1+t^2} dt$$ So we have simple poles at $i$,$-i$. Why does using a quarter of a circle in ...
25
votes
8answers
5k views

When two functions are equal, but not.

I haven't looked into it much, but this is something I've been aware of that I know I need to look into. When I have a function $f(x)=\frac{x+1}{x+1}$, There is a discontinuity at $x=-1$, yet $\frac{...
3
votes
2answers
225 views

Evaluating $\int\sqrt{150^2-x^2} \cdot dx$

I'm studying for my finals and I have this integral that I'm trying to evaluate (part of a bigger problem): $$\int\sqrt{150^2-x^2} \cdot dx$$ I have evaluated a few integrals of this type before so ...
2
votes
3answers
233 views

Solving $\int\frac{\ln(1+e^x)}{e^x} \space dx$

I'm trying to solve this integral. $$\int\frac{\ln(1+e^x)}{e^x} \space dx$$ I try to solve it using partial integration twice, but then I get to this point (where $t = e^x$ and $dx = \frac{1}{t} dt$)...
2
votes
2answers
2k views

Is this true about integrating composite functions?

Let's say that I'm integrating a composite function, say $f(g(x))$, that is in a form to which I can apply the substitution rule. Is it true to say that both $f$ and $g$ must be differentiable? I ...
11
votes
0answers
1k views

Cauchy-Formula for Repeated Lebesgue-Integration

Recently, I came across the following statements. They were annotated as consequences of Fubini's Theorem but neither proof nor reference were given. Let $f:[a,b]\times [a,b]\to\mathbb{R}$ be ...