All aspects of integration, including the definition of the integral and computing different types of integrals. For questions solely about the properties of integrals, don't use this tag alone! Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another ...

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16
votes
2answers
396 views
+100

Closed form for the integral $\int_{0}^{\infty}\frac{\ln^{2}(x)\ln(1+x)}{(1-x)(x^{2}+1)}dx$

Here is a challenging one maybe some would like a go at. Show that: ...
5
votes
0answers
105 views
+150

Separability of a set with norm $\thickapprox$ $L^1$ +$L^{\infty}$

Let $(M, \mathcal{A}, \mu)$ a complete separable probability space. Recall that complete means that any subset of a measurable set with zero measure is measure (and has zero measure) and separable ...
11
votes
0answers
159 views
+50

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} ...
1
vote
1answer
86 views
+100

Surjectivity of an integration map

N.B.: Thanks to studiosus answer I realised I should ask for more conditions or otherwise the answer is straightforwardly wrong. I rechecked my problem and added new assumptions that I boldface. ...
0
votes
0answers
43 views
+50

Countable and uncountable sets in Riemann integration

The Riemann integral over $[a,b]$ of a continuous function $f$ is $$\int\limits_a^bf(x)dx=\lim\limits_{\delta\rightarrow 0} \sum\limits_{i=0}^{n-1} (x_{i+1}-x_i)f(c_i)$$ where $c_i\in[x_i,x_{i+1}]$ ...