# 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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### 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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### 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 ...
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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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### 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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### 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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### 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 ...
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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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### 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 ...
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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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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 $\... 128 votes 2 answers 5k views ### How to prove$\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 $$\int_0^1 \tan^{-1}\left[\frac{\tanh^{-1}x-\tan^{-1}x}{\pi+\tanh^{-1}x-\tan^{-1}x}\right]\frac{dx}{x}=\frac{\pi}{8}\ln\frac{\pi^2}{8}?$$ 122 votes 5 answers 9k views ### 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 ... 116 votes 10 answers 12k views ### 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 ... 115 votes 3 answers 12k views ### 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]\, ... 113 votes 9 answers 13k views ### 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 ... 109 votes 12 answers 14k views ### 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? $$\int_0^\infty\frac{\cos x}{1+x^2}\,\mathrm{d}x$$ 108 votes 8 answers 45k views ### 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 ... 107 votes 1 answer 4k views ### 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 $$\frac1N\sum_k x_k ... 106 votes 11 answers 8k views ### Closed form for \int_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx } I've been looking at$$\int\limits_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$$It seems that it always evaluates in terms of \sin X and \pi, where X is to be determined. For example:$$\... 105 votes 3 answers 27k views ###$\int_{-\infty}^{+\infty} e^{-x^2} dx$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 ... 103 votes 14 answers 10k views ### 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 ... 101 votes 7 answers 7k views ### 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 $$\lim_{x\to c}\frac{f'(x)}{g'(x)}= \lim_{... 100 votes 4 answers 19k views ### An integral involving Airy functions \int_0^\infty\frac{x^p}{\operatorname{Ai}^2 x + \operatorname{Bi}^2 x}\mathrm dx I need your help with this integral:$$\mathcal{K}(p)=\int_0^\infty\frac{x^p}{\operatorname{Ai}^2 x + \operatorname{Bi}^2 x}\mathrm dx,$$where \operatorname{Ai}, \operatorname{Bi} are Airy ... 98 votes 8 answers 39k views ### Evaluate the integral: \int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm dx Compute$$\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm dx$$98 votes 6 answers 115k views ### Why is the area under a curve the integral? I understand how derivatives work based on the definition, and the fact that my professor explained it step by step until the point where I can derive it myself. However when it comes to the area ... 93 votes 3 answers 9k views ### How to evaluate \int_{0}^{\infty} \frac{x^{-\mathfrak{i}a}}{x^2+bx+1} \,\mathrm{d}x using complex analysis? We were told today by our teacher (I suppose to scare us) that in certain schools for physics in Soviet Russia there was as an entry examination the following integral given$$\int\limits_{0}^{\... 92 votes 11 answers 17k views ### 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 ... 90 votes 5 answers 10k views ### 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 ... 90 votes 2 answers 6k views ### Conjecture$\int_0^1\frac{\mathrm dx}{\sqrt{1-x}\ \sqrtx\ \sqrt{2-x\,\sqrt3}}\stackrel?=\frac{2\,\sqrt2}{3\,\sqrt3}\pi$$$\int_0^1\frac{\mathrm dx}{\sqrt{1-x}\ \sqrtx\ \sqrt{2-x\,\sqrt3}}\stackrel?=\frac{2\,\sqrt2}{3\,\sqrt3}\pi\tag1$$ The equality numerically holds up to at least$10^4$decimal digits. Can ... 87 votes 8 answers 9k views ### 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 ... 87 votes 15 answers 25k views ### 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 $$\int_0^\infty \frac{\sin x}{x} \mathrm ... 86 votes 5 answers 6k views ### Calculating the volume of a restaurant take-away box that is circular on the bottom and square on the top Having a bit of a problem calculating the volume of a take-away box: I originally wanted to use integration to measure it by rotating around the x-axiz, but realised that when folded the top becomes ... 86 votes 2 answers 5k views ### Conjecture \int_0^1\frac{dx}{\sqrtx\,\sqrt{1-x}\,\sqrt{1-x\left(\sqrt{6}\sqrt{12+7\sqrt3}-3\sqrt3-6\right)^2}}=\frac\pi9(3+\sqrt2\sqrt{27}) Let$$\alpha=\sqrt{6}\ \sqrt{12+7\,\sqrt3}-3\,\sqrt3-6\,=\,\big(2+\sqrt{3}\big) \big(\sqrt{2} \sqrt{27}-3\big)\,=\,\frac{3\sqrt{3}}{3+\sqrt2\ \sqrt{27}}.\tag1$$Note that \alpha is the unique ... 81 votes 8 answers 11k views ### What is integration by parts, really? Integration by parts comes up a lot - for instance, it appears in the definition of a weak derivative / distributional derivative, or as a tool that one can use to turn information about higher ... 80 votes 4 answers 4k views ### Integrals of \sqrt{x+\sqrt{\phantom|\dots+\sqrt{x+1}}} in elementary functions Let f_n(x) be recursively defined as$$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 votes 2 answers 4k views ### Ramanujan log-trigonometric integrals I discovered the following conjectured identity numerically while studying a family of related integrals. Let's set $$R^{+}:= \frac{2}{\pi}\int_{0}^{\pi/2}\sqrt[\normalsize{8}]{x^2 + \ln^2\!\cos x} ... 79 votes 4 answers 3k views ### Integral \int_1^\infty\operatorname{arccot}\left(1+\frac{2\pi}{\operatorname{arcoth}x-\operatorname{arccsc}x}\right)\frac{\mathrm dx}{\sqrt{x^2-1}} Consider the following integral:$$\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}}\,,$$... 78 votes 18 answers 37k views ### 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 ... 77 votes 4 answers 5k views ### 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 ...