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dustin
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510 votes
10 answers
260k views

Integral $\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right) \mathrm dx$

490 votes
10 answers
64k views

Best Sets of Lecture Notes and Articles

  • 48.9k
264 votes
5 answers
21k views

Evaluate $ \int_{0}^{\frac{\pi}2}\frac1{(1+x^2)(1+\tan x)}\:\mathrm dx$

  • 50.2k
191 votes
28 answers
128k views

What is a good complex analysis textbook, barring Ahlfors's?

  • 1,185
166 votes
9 answers
40k views

Lesser-known integration tricks

131 votes
10 answers
29k views

How to find ${\large\int}_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx$

117 votes
10 answers
12k views

Find the average of $\sin^{100} (x)$ in 5 minutes?

116 votes
17 answers
49k views

Nice proofs of $\zeta(4) = \frac{\pi^4}{90}$?

  • 53.2k
106 votes
11 answers
8k views

Closed form for $ \int_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$

  • 117k
85 votes
25 answers
13k views

Easy example why complex numbers are cool

  • 1,734
82 votes
1 answer
4k views

The closed form of $\int_0^{\pi/4}\frac{\log(1-x) \tan^2(x)}{1-x\tan^2(x)} \ dx$

  • 43.3k
75 votes
9 answers
11k views

Evaluating $\int_0^\infty \sin x^2\, dx$ with real methods?

  • 24.4k
73 votes
3 answers
42k views

An Explanation of the Kalman Filter

  • 147
69 votes
4 answers
3k views

Show that $\int_{0}^{\pi/2}\frac {\log^2\sin x\log^2\cos x}{\cos x\sin x}\mathrm{d}x=\frac14\left( 2\zeta (5)-\zeta(2)\zeta (3)\right)$

  • 3,805
56 votes
3 answers
4k views

Counterexample Math Books

  • 55.2k
51 votes
3 answers
3k views

Prove $\int\limits_{0}^{\pi/2}\frac{dx}{1+\sin^2{(\tan{x})}}=\frac{\pi}{2\sqrt{2}}\bigl(\frac{e^2+3-2\sqrt{2}}{e^2-3+2\sqrt{2}}\bigr)$

46 votes
8 answers
3k views

Closed-form of $\int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx $

  • 11.7k
44 votes
5 answers
3k views

Prove $\int_0^{\infty}\! \frac{\mathbb{d}x}{1+x^n}=\frac{\pi}{n \sin\frac{\pi}{n}}$ using real analysis techniques only

  • 1,408
42 votes
9 answers
20k views

Computing the integral of $\log(\sin x)$

  • 465
42 votes
6 answers
2k views

Computing $ \int_0^\infty \frac{\log x}{\exp x} \ dx $ [duplicate]

  • 2,710
39 votes
3 answers
2k views

Closed form for $\int_0^\infty\arctan\Bigl(\frac{2\pi}{x-\ln\,x+\ln(\frac\pi2)}\Bigr)\frac{dx}{x+1}$ [closed]

39 votes
4 answers
2k views

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

  • 13.6k
37 votes
3 answers
2k views

Interesting log sine integrals $\int_0^{\pi/3} \log^2 \left(2\sin \frac{x}{2} \right)dx= \frac{7\pi^3}{108}$

  • 1,441
33 votes
7 answers
12k views

Computing $\int_{0}^{\pi}\ln\left(1-2a\cos x+a^2\right) \, dx$

33 votes
6 answers
85k views

What books are recommended for learning calculus on my own? [duplicate]

  • 431
31 votes
4 answers
2k views

Evaluate $\int_0^\infty \frac{\log(1+x^3)}{(1+x^2)^2}dx$ and $\int_0^\infty \frac{\log(1+x^4)}{(1+x^2)^2}dx$

31 votes
2 answers
6k views

Connection between the Laplace transform and generating functions

  • 5,451
30 votes
5 answers
959 views

Closed form for $\int_{-\infty}^\infty\operatorname{sech}(x)\operatorname{sech}(a\, x)\ dx$

29 votes
3 answers
1k views

How to find PV $\int_0^\infty \frac{\log \cos^2 \alpha x}{\beta^2-x^2} \, \mathrm dx=\alpha \pi$

  • 9,452
28 votes
1 answer
1k views

Integrate $ \int_{0}^{\frac{\pi}{4}}\tan^{-1}\left(\frac{\sqrt{2}\cos3 \phi}{\left(2\cos 2 \phi+ 3\right)\sqrt{\cos 2 \phi}}\right)d\phi$

  • 6,560