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• Comment: More context about the subject is desirable. Right now the article simply states the equation and provides little else. More sources are also desirable. Rambley ^{(talk / contribs)} 11:45, 3 December 2025 (UTC)

In mathematics, the Serret's integral is a definite integral named after French mathematician Joseph-Alfred Serret. It takes the form of

${\displaystyle \displaystyle \int _{0}^{1}{\frac {\ln(x+1)}{x^{2}+1}}\,dx={\pi \over 8}\ln 2.\approx 0.272198...}$

It is named after Joseph-Alfred Serret because of an note^[1] published in 1844 on the Journal de mathématiques pures et appliquées.

We set

${\displaystyle x=\tan \varphi ,\qquad {\frac {dx}{1+x^{2}}}=d\varphi }$

So the integral becomes

${\displaystyle \int _{0}^{1}{\frac {\log(1+x)}{1+x^{2}}}\,dx=\int _{0}^{\pi /4}\log(1+\tan \varphi )\,d\varphi }$

Now,

${\displaystyle 1+\tan \varphi ={\frac {{\sqrt {2}}\,\cos \!\left({\tfrac {\pi }{4}}-\varphi \right)}{\cos \varphi }}}$

and therefore

${\displaystyle \ln(1+\tan \varphi )={\tfrac {1}{2}}\ln 2+\ln \!\left(\cos \!\left({\tfrac {\pi }{4}}-\varphi \right)\right)-\ln(\cos \varphi ).}$

Thus,

${\displaystyle \int _{0}^{\pi /4}\ln(1+\tan \varphi )\,d\varphi ={\frac {\pi }{8}}\ln 2+\int _{0}^{\pi /4}\ln \!\left(\cos \!\left({\tfrac {\pi }{4}}-\varphi \right)\right)\,d\varphi -\int _{0}^{\pi /4}\ln(\cos \varphi )\,d\varphi }$

Since the two integrals cancel each other

${\displaystyle \int _{0}^{\pi /4}\ln(\cos \!\left({\tfrac {\pi }{4}}-\varphi \right))\,d\varphi =\int _{0}^{\pi /4}\ln(\cos \varphi )\,d\varphi }$

We finally get the result

${\displaystyle \int _{0}^{1}{\frac {\ln(1+x)}{1+x^{2}}}\,dx={\frac {\pi }{8}}\ln 2}$.

In the Gradshteyn & Ryzhik along with original integral, there are notable generalizations.

• I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed., Academic Press (2000), eqn. 4.291.8.
• G. Boros, Victor Hugo Moll, Irresistible Integrals (2004), pp. 243
• G. E. Raynor, On Serret's Integral Formula (PDF; 423  kB) (1939)
• Sloane, N. J. A. The On-Line Encyclopedia of Integer Sequences, Sequence n. A102886