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 Post subject: Nice Integral (Putnam)
PostPosted: Sat, 4 Sep 2010 04:57:36 UTC 
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Evaluate

$\int_0^1 \frac{\ln{(x+1)}}{x^2+1}dx


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PostPosted: Sat, 4 Sep 2010 06:20:07 UTC 
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qwirk wrote:
Evaluate

$\int_0^1 \frac{\ln{(x+1)}}{x^2+1}dx


It is easy with contour. Can you do without?

_________________
\begin{aligned}
Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\
Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\
Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\
Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\
Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\
Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4
\end{aligned}


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PostPosted: Sat, 4 Sep 2010 10:34:53 UTC 
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outermeasure wrote:
qwirk wrote:
Evaluate

$\int_0^1 \frac{\ln{(x+1)}}{x^2+1}dx


It is easy with contour. Can you do without?


I use
Spoiler:
$ x=\tan \theta$


One of the few Putnam's I have managed to nut out...


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