Questions tagged [integration]
For questions about the properties of integrals. Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that describe the type of integral being considered. This tag often goes along with the (calculus) tag.
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Integral $\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right) \mathrm dx$
I need help with this integral:
$$I=\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right)\ \mathrm dx.$$
The integrand graph looks like this:
$\hspace{1in}$
The ...
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answers
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Calculating the length of the paper on a toilet paper roll
Fun with Math time.
My mom gave me a roll of toilet paper to put it in the bathroom, and looking at it I immediately wondered about this: is it possible, through very simple math, to calculate (with ...
348
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answers
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How can you prove that a function has no closed form integral?
In the past, I've come across statements along the lines of "function $f(x)$ has no closed form integral", which I assume means that there is no combination of the operations:
addition/...
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21
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Really advanced techniques of integration (definite or indefinite)
Okay, so everyone knows the usual methods of solving integrals, namely u-substitution, integration by parts, partial fractions, trig substitutions, and reduction formulas. But what else is there? ...
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Evaluate $ \int_{0}^{\frac{\pi}2}\frac1{(1+x^2)(1+\tan x)}\:\mathrm dx$
Evaluate the following integral
$$
\tag1\int_{0}^{\frac{\pi}{2}}\frac1{(1+x^2)(1+\tan x)}\,\mathrm dx
$$
My Attempt:
Letting $x=\frac{\pi}{2}-x$ and using the property that
$$
\int_{0}^{a}f(x)\,\...
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The Integral that Stumped Feynman?
In "Surely You're Joking, Mr. Feynman!," Nobel-prize winning Physicist Richard Feynman said that he challenged his colleagues to give him an integral that they could evaluate with only ...
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"Integral Milking"
I begin this post with a plea: please don't be too harsh with this post for being off topic or vague. It's a question about something I find myself doing as a mathematician, and wonder whether others ...
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Is this really a categorical approach to integration?
Here's an article by Reinhard Börger I found recently whose title and content, prima facie, seem quite exciting to me, given my misadventures lately (like this and this); it's called, "A ...
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Evaluation of Gaussian integral $\int_{0}^{\infty} \mathrm{e}^{-x^2} dx$
How to prove
$$\int_{0}^{\infty} \mathrm{e}^{-x^2}\, dx = \frac{\sqrt \pi}{2}$$
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Generalizing $\int_{0}^{1} \frac{\arctan\sqrt{x^{2} + 2}}{\sqrt{x^{2} + 2}} \, \frac{\operatorname dx}{x^{2}+1} = \frac{5\pi^{2}}{96}$
The following integral
\begin{align*}
\int_{0}^{1} \frac{\arctan\sqrt{x^{2} + 2}}{\sqrt{x^{2} + 2}} \, \frac{dx}{x^{2}+1} = \frac{5\pi^{2}}{96}
\tag{1}
\end{align*}
is called the Ahmed's integral ...
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Some users are mind bogglingly skilled at integration. How did they get there?
Looking through old problems, it is not difficult to see that some users are beyond incredible at computing integrals. It only took a couple seconds to dig up an example like this.
Especially in a ...
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A math contest problem $\int_0^1\ln\left(1+\frac{\ln^2x}{4\,\pi^2}\right)\frac{\ln(1-x)}x \ \mathrm dx$
A friend of mine sent me a math contest problem that I am not able to solve (he does not know a solution either). So, I thought I might ask you for help.
Prove:
$$\int_0^1\ln\left(1+\frac{\ln^2x}{...
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Symmetry of function defined by integral
Define a function $f(\alpha, \beta)$, $\alpha \in (-1,1)$, $\beta \in (-1,1)$ as
$$ f(\alpha, \beta) = \int_0^{\infty} dx \: \frac{x^{\alpha}}{1+2 x \cos{(\pi \beta)} + x^2}$$
One can use, for ...
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Evaluate $\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}\mathrm dx$
I am trying to find a closed form for
$$\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}\mathrm dx = 0.094561677526995723016 \cdots$$
It seems that the answer is
$$\frac{\pi^2}{12}\left( 1-\...
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8
answers
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Lesser-known integration tricks
I am currently studying for the GRE math subject test, which heavily tests calculus. I've reviewed most of the basic calculus techniques (integration by parts, trig substitutions, etc.) I am now ...
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answers
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Striking applications of integration by parts
What are your favorite applications of integration by parts?
(The answers can be as lowbrow or highbrow as you wish. I'd just like to get a bunch of these in one place!)
Thanks for your ...
166
votes
42
answers
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Is there any integral for the Golden Ratio?
I was wondering about important/famous mathematical constants, like $e$, $\pi$, $\gamma$, and obviously the golden ratio $\phi$.
The first three ones are really well known, and there are lots of ...
132
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answers
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Why is integration so much harder than differentiation?
If a function is a combination of other functions whose derivatives are known via composition, addition, etc., the derivative can be calculated using the chain rule and the like. But even the product ...
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votes
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answers
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How to find ${\large\int}_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx$
Please help me to find a closed form for this integral:
$$I=\int_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx\tag1$$
I suspect it might exist because there are similar integrals having closed forms:
$$\begin{...
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Is there an integral that proves $\pi > 333/106$?
The following integral,
$$ \int_0^1 \frac{x^4(1-x)^4}{x^2 + 1} \mathrm{d}x = \frac{22}{7} - \pi $$
is clearly positive, which proves that $\pi < 22/7$.
Is there a similar integral which proves $\...
anon
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2
answers
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How to prove $\int_0^1\tan^{-1}\left[\frac{\tanh^{-1}x-\tan^{-1}x}{\pi+\tanh^{-1}x-\tan^{-1}x}\right]\frac{dx}{x}=\frac{\pi}{8}\ln\frac{\pi^2}{8}?$
How can one prove that
$$\int_0^1 \tan^{-1}\left[\frac{\tanh^{-1}x-\tan^{-1}x}{\pi+\tanh^{-1}x-\tan^{-1}x}\right]\frac{dx}{x}=\frac{\pi}{8}\ln\frac{\pi^2}{8}?$$
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Help find hard integrals that evaluate to $59$?
My father and I, on birthday cards, give mathematical equations for each others new age. This year, my father will be turning $59$.
I want to try and make a definite integral that equals $59$. So ...
120
votes
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answers
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Find the average of $\sin^{100} (x)$ in 5 minutes?
I read this quote attributed to VI Arnold.
"Who can't calculate the average value of the one hundredth power of the sine function within five minutes, doesn't understand mathematics - even if he ...
Please Delete Account
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Compute $\int_0^{\pi/4}\frac{(1-x^2)\ln(1+x^2)+(1+x^2)-(1-x^2)\ln(1-x^2)}{(1-x^4)(1+x^2)} x\exp(\frac{x^2-1}{x^2+1}) dx$
Compute the following integral
\begin{equation}
\int_0^{\Large\frac{\pi}{4}}\left[\frac{(1-x^2)\ln(1+x^2)+(1+x^2)-(1-x^2)\ln(1-x^2)}{(1-x^4)(1+x^2)}\right] x\, \exp\left[\frac{x^2-1}{x^2+1}\right]\, ...
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answers
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Proof of Frullani's theorem
How can I prove the Theorem of Frullani? I did not even know all the hypothesis that $f$ must satisfy, but I think that this are
Let $\,f:\left[ {0,\infty } \right) \to \mathbb R$ be a a continuously ...
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Lebesgue integral basics
I'm having trouble finding a good explanation of the Lebesgue integral. As per the definition, it is the expectation of a random variable. Then how does it model the area under the curve? Let's take ...
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Calculating the integral $\int_0^\infty \frac{\cos x}{1+x^2}\, \mathrm{d}x$ without using complex analysis
Suppose that we do not know anything about the complex analysis (numbers). In this case, how to calculate the following integral in closed form?
$$\int_0^\infty\frac{\cos x}{1+x^2}\,\mathrm{d}x$$
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Closed form for $ \int_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$
I've been looking at
$$\int\limits_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$$
It seems that it always evaluates in terms of $\sin X$ and $\pi$, where $X$ is to be determined. For example:
$$\...
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$\int_{-\infty}^{+\infty} e^{-x^2} dx$ with complex analysis
Inspired by this recently closed question, I'm curious whether there's a way to do the Gaussian integral using techniques in complex analysis such as contour integrals.
I am aware of the calculation ...
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Is this continuous analogue to the AM–GM inequality true?
First let us remind ourselves of the statement of the AM–GM inequality:
Theorem: (AM–GM Inequality) For any sequence $(x_n)$ of $N\geqslant 1$ non-negative real numbers, we have $$\frac1N\sum_k x_k ...
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Ways to evaluate $\int \sec \theta \, \mathrm d \theta$
The standard approach for showing $\int \sec \theta \, \mathrm d \theta = \ln|\sec \theta + \tan \theta| + C$ is to multiply by $\dfrac{\sec \theta + \tan \theta}{\sec \theta + \tan \theta}$ and then ...
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An integral involving Airy functions $\int_0^\infty\frac{x^p}{\operatorname{Ai}^2 x + \operatorname{Bi}^2 x}\mathrm dx$
I need your help with this integral:
$$\mathcal{K}(p)=\int_0^\infty\frac{x^p}{\operatorname{Ai}^2 x + \operatorname{Bi}^2 x}\mathrm dx,$$
where $\operatorname{Ai}$, $\operatorname{Bi}$ are Airy ...
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Evaluate the integral: $\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm dx$
Compute
$$\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm dx$$
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Why is the area under a curve the integral?
I understand how derivatives work based on the definition, and the fact that my professor explained it step by step until the point where I can derive it myself.
However when it comes to the area ...
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Does L'Hôpital's work the other way?
As referred in Wikipedia (see the specified criteria there), L'Hôpital's rule says,
$$
\lim_{x\to c}\frac{f(x)}{g(x)}=\lim_{x\to c}\frac{f'(x)}{g'(x)}
$$
As
$$
\lim_{x\to c}\frac{f'(x)}{g'(x)}=
\lim_{...
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How to evaluate $\int_{0}^{\infty} \frac{x^{-\mathfrak{i}a}}{x^2+bx+1} \,\mathrm{d}x$ using complex analysis?
We were told today by our teacher (I suppose to scare us) that in certain schools for physics in Soviet Russia there was as an entry examination the following integral given
$$\int\limits_{0}^{\...
user505183
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Demystify integration of $\int \frac{1}{x} \mathrm dx$
I've learned in my analysis class, that
$$ \int \frac{1}{x} \mathrm dx = \ln(x). $$
I can live with that, and it's what I use when solving equations like that.
But how can I solve this, without ...
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Why are gauge integrals not more popular?
A recent answer reminded me of the gauge integral, which you can read about here.
It seems like the gauge integral is more general than the Lebesgue integral, e.g. if a function is Lebesgue ...
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Conjecture $\int_0^1\frac{\mathrm dx}{\sqrt{1-x}\ \sqrt[4]x\ \sqrt[4]{2-x\,\sqrt3}}\stackrel?=\frac{2\,\sqrt2}{3\,\sqrt[8]3}\pi$
$$\int_0^1\frac{\mathrm dx}{\sqrt{1-x}\ \sqrt[4]x\ \sqrt[4]{2-x\,\sqrt3}}\stackrel?=\frac{2\,\sqrt2}{3\,\sqrt[8]3}\pi\tag1$$
The equality numerically holds up to at least $10^4$ decimal digits.
Can ...
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Compute $\int \frac{\sin(x)}{\sin(x)+\cos(x)}\mathrm dx$
I'm having trouble computing the integral:
$$\int \frac{\sin(x)}{\sin(x)+\cos(x)}\mathrm dx.$$
I hope that it can be expressed in terms of elementary functions. I've tried simple substitutions such as ...
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15
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Proof of $\int_0^\infty \left(\frac{\sin x}{x}\right)^2 \mathrm dx=\frac{\pi}{2}.$
I am looking for a short proof that $$\int_0^\infty \left(\frac{\sin x}{x}\right)^2 \mathrm dx=\frac{\pi}{2}.$$
What do you think?
It is kind of amazing that $$\int_0^\infty \frac{\sin x}{x} \mathrm ...
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Conjecture $\int_0^1\frac{dx}{\sqrt[3]x\,\sqrt[6]{1-x}\,\sqrt{1-x\left(\sqrt{6}\sqrt{12+7\sqrt3}-3\sqrt3-6\right)^2}}=\frac\pi9(3+\sqrt2\sqrt[4]{27})$
Let
$$\alpha=\sqrt{6}\ \sqrt{12+7\,\sqrt3}-3\,\sqrt3-6\,=\,\big(2+\sqrt{3}\big) \big(\sqrt{2} \sqrt[4]{27}-3\big)\,=\,\frac{3\sqrt{3}}{3+\sqrt2\ \sqrt[4]{27}}.\tag1$$
Note that $\alpha$ is the unique ...
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Calculating the volume of a restaurant take-away box that is circular on the bottom and square on the top
Having a bit of a problem calculating the volume of a take-away box:
I originally wanted to use integration to measure it by rotating around the x-axiz, but realised that when folded the top becomes ...
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Can a limit of an integral be moved inside the integral?
After coming across this question: How to verify this limit, I have the following question:
When taking the limit of an integral, is it valid to move the limit inside the integral, providing the ...
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What is integration by parts, really?
Integration by parts comes up a lot - for instance, it appears in the definition of a weak derivative / distributional derivative, or as a tool that one can use to turn information about higher ...
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A strange integral: $\int_{-\infty}^{+\infty} {dx \over 1 + \left(x + \tan x\right)^2} = \pi.$
While browsing on Integral and Series, I found a strange integral posted by @Sangchul Lee. His post doesn't have a response for more than a month, so I decide to post it here. I hope he doesn't mind ...
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votes
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Ramanujan log-trigonometric integrals
I discovered the following conjectured identity numerically while studying a family of related integrals.
Let's set
$$
R^{+}:= \frac{2}{\pi}\int_{0}^{\pi/2}\sqrt[\normalsize{8}]{x^2 + \ln^2\!\cos x}
...
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votes
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Integrals of $\sqrt{x+\sqrt{\phantom|\dots+\sqrt{x+1}}}$ in elementary functions
Let $f_n(x)$ be recursively defined as
$$f_0(x)=1,\ \ \ f_{n+1}(x)=\sqrt{x+f_n(x)},\tag1$$
i.e. $f_n(x)$ contains $n$ radicals and $n$ occurences of $x$:
$$f_1(x)=\sqrt{x+1},\ \ \ f_2(x)=\sqrt{x+\sqrt{...
79
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18
answers
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List of interesting integrals for early calculus students
I am teaching Calc 1 right now and I want to give my students more interesting examples of integrals. By interesting, I mean ones that are challenging, not as straightforward (though not extremely ...
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Integral $\int_1^\infty\operatorname{arccot}\left(1+\frac{2\pi}{\operatorname{arcoth}x-\operatorname{arccsc}x}\right)\frac{\mathrm dx}{\sqrt{x^2-1}}$
Consider the following integral:
$$\mathcal{I}=\int_1^\infty\operatorname{arccot}\left(1+\frac{2\,\pi}{\operatorname{arcoth}x\,-\,\operatorname{arccsc}x}\right)\frac{\mathrm dx}{\sqrt{x^2-1}}\,,$$
...