# 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 whose upper limit is the integral itself: $\int_{0}^{\int_{0}^{\ldots}\frac{1}{\sqrt{x}} \ \mathrm{d}x} \frac{1}{\sqrt{x}} \ \mathrm{d}x$

I recently encountered the following definite integral: $$\int_0^{\int_0^\ldots \frac{1}{\sqrt{x}} \ \mathrm{d}x} \frac{1}{\sqrt{x}} \ \mathrm{d}x$$ where "$\ldots$" seems to indicate that the upper ...
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### Did I just discover this integration formula?

One night, I discovered an integration relationship. That relationship allows to quickly integrate squares of functions (and even more, but I will talk about this at the end). I was wondering if ...
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### Can I solve an integral (or other tough problem) by playing with knots?

I've seen that in calculating things in knot theory that involves a lot of hard looking integrals and matrices, even though the knots themselves appear fairly simple. So is there some way in which ...
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### Weber-type integral

In connection with this answer, I came across the following integral: $$\int_{0}^{\infty} \frac{du}{u} \: \,e^{-t u^2} \frac{J_0(u) Y_0(r u)-J_0(r u) Y_0(u)}{J_0^2(u)+Y_0^2(u)}$$ where $r \gt 1$. I ...
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### Evaluating $\int_0^1 \log \log \left(\frac{1}{x}\right) \frac{dx}{1+x^2}$

Show that $\displaystyle{\int_0^1 \log \log \left(\frac{1}{x}\right) \frac{dx}{1+x^2} = \frac{\pi}{2}\log \left(\sqrt{2\pi} \Gamma\left(\frac{3}{4}\right) / \Gamma\left(\frac{1}{4}\right)\right)}$ ...
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### $\int_{0}^{\infty} \frac{\cos x - e^{-x^2}}{x} \ dx$ Evaluate Integral

Evaluate $$\int_{0}^{\infty} \frac{\cos x - e^{-x^2}}{x} \ dx$$
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### Prove $\int_0^1\frac{x^2-2\,x+2\ln(1+x)}{x^3\,\sqrt{1-x^2}}\mathrm dx=\frac{\pi^2}8-\frac12$

How can I prove the following identity? $$\int_0^1\frac{x^2-2\,x+2\ln(1+x)}{x^3\,\sqrt{1-x^2}}\mathrm dx=\frac{\pi^2}8-\frac12$$
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### What is to geometric mean as integration is to arithmetic mean?

The arithmetic mean of $y_i \ldots y_n$ is: $$\frac{1}{n}\sum_{i=1}^n~y_i$$ For a smooth function $f(x)$, we can find the arithmetic mean of $f(x)$ from $x_0$ to $x_1$ by taking $n$ samples and ...
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### Closed form for $\int_0^1\frac{x^{5/6}}{(1-x)^{1/6}\,(1+2\,x)^{4/3}}\log\left(\frac{1+2x}{x\,(1-x)}\right)\,dx$

I need to evaluate this integral: $$Q=\int_0^1\frac{x^{5/6}}{(1-x)^{1/6}\,(1+2\,x)^{4/3}}\log\left(\frac{1+2x}{x\,(1-x)}\right)\,dx.$$ I tried it in Mathematica, but it was not able to find a closed ...
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### Surely You're Joking, Mr. Feynman! $\int_0^\infty\frac{\sin^2x}{x^2(1+x^2)}\,dx$ [duplicate]

Prove the following $$\int_0^\infty\frac{\sin^2x}{x^2(1+x^2)}\,dx=\frac{\pi}{4}+\frac{\pi}{4e^2}$$ I would love to see how Mathematics SE users prove the integral ...
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### Evaluating $\int_0^{\infty}\frac{\ln(x^2+1)}{x^2+1}dx$

How would I go about evaluating this integral? $$\int_0^{\infty}\frac{\ln(x^2+1)}{x^2+1}dx.$$ What I've tried so far: I tried a semicircular integral in the positive imaginary part of the complex ...
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### Why does the fundamental theorem of calculus work?

I've known for some time that one of the fundamental theorems of calculus states: $$\int_{a}^{b}\ f'(x){\mathrm{d} x} = f(b)-f(a)$$ Despite using this formula, I've yet to see a proof or even a ...
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### Integral $\int_0^1\frac{1-x^2+\left(1+x^2\right)\ln x}{\left(x+x^2\right)\ln^3x}dx$

I'm struggling with this integral $$I=\int_0^1\frac{1-x^2+\left(1+x^2\right)\ln x}{\left(x+x^2\right)\ln^3x}dx.\tag1$$ Mathematica could not evaluate it in a closed form. Its numeric value is ...
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### A closed form for $\int_0^\infty\frac{\ln(x+4)}{\sqrt{x\,(x+3)\,(x+4)}}dx$

I need to a evaluate the following integral $$I=\int_0^\infty\frac{\ln(x+4)}{\sqrt{x\,(x+3)\,(x+4)}}dx.$$ Both Mathematica and Maple failed to evaluate it in a closed form, and lookups of the ...
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### Integral ${\large\int}_0^\infty\frac{dx}{\sqrt[4]{7+\cosh x}}$

How to prove the following conjectured identity? $$\int_0^\infty\frac{dx}{\sqrt[4]{7+\cosh x}}\stackrel{\color{#a0a0a0}?}=\frac{\sqrt[4]6}{3\sqrt\pi}\Gamma^2\big(\tfrac14\big)\tag1$$ It holds ...
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### Good book on evaluating difficult definite integrals (without elementary antiderivatives)?

I am very interested in evaluating difficult definite integrals without elementary antiderivatives by manipulating the integral somehow (e.g. contour integration, interchanging order of integration/...
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### Evaluating $\int_0^1 \frac{\log x \log \left(1-x^4 \right)}{1+x^2}dx$

I am trying to prove that $$\int_0^1 \frac{\log x \log \left(1-x^4 \right)}{1+x^2}dx = \frac{\pi^3}{16}-3G\log 2 \tag{1}$$ where $G$ is Catalan's Constant. I was able to express it in terms of ...
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### Generalizing the trick for integrating $\int_{-\infty}^\infty e^{-x^2}\mathrm dx$?

There is a well-known trick for integrating $\int_{-\infty}^\infty e^{-x^2}\mathrm dx$, which is to write it as $\sqrt{\int_{-\infty}^\infty e^{-x^2}\mathrm dx\int_{-\infty}^\infty e^{-y^2}\mathrm dy}$...
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### Integral $\int_0^1\frac{\ln x}{\left(1+x\right)\left(1+x^{-\left(2+\sqrt3\right)}\right)}dx$
There is a curious known integral: \int_0^1\frac{\ln\left(1+x^{2+\sqrt{3\vphantom{\large3}}}\right)}{1+x}dx=\frac{\pi^2}{12}\left(1-\sqrt{3\vphantom{\large3}}\right)+\ln \left(1+\sqrt{3\vphantom{\...
### On calculating $\int_0^1\ln(1-x^2)\;{\mathrm dx}$ — where is the mistake?
I've seen the integral $\displaystyle \int_0^1\ln(1-x^2)\;{dx}$ on a thread in this forum and I tried to calculate it by using power series. I wrote the integral as a sum then again as an integral. ...