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Joshua King came to Cambridge from Hawkshead Grammar School. It was soon evident that the school had produced someone of importance. He became Senior Wrangler, and his reputation in Cambridge was immense. It was believed that nothing less than a Second Newton had appeared. They expected his work as a mathematician to make an epoch in the science. At an early age he became President of Queens’; later, he was Lucasian Professor. He published nothing; in fact, he did no mathematical work. But as long as he kept his health, he was an active and prominent figure in Cambridge, and he maintained his enormous reputation. When he died, it was felt that the memory of such an extraordinary man should not be permitted to die out, and that his papers should be published. So his papers were examined, and nothing whatever worth publishing was found.


5h
answered Integration without complex analysis on rational-improper integral
1d
comment Evaluate $\int \ln(1 + e^x)\ \mathrm dx$
See Liouville's theorem and the Risch algorithm.
1d
comment Convergence of a integral: $\int_{0}^{1} |\ln (x)|^n \ dx$
This is Euler's first integral expression for the $\Gamma$ function. It evaluates to $n!$ for $n>-1$.
1d
answered The sequence of improper integrals of the form $\int\frac{dx}{1+x^{2n}}$
1d
comment The sequence of improper integrals of the form $\int\frac{dx}{1+x^{2n}}$
To simplify $I_6$, use Euler's reflection formula for the $\Gamma$ function.
1d
answered How to calculate variant of geometric series based on sequences of Catalan numbers?
1d
comment Evaluation of $\int \frac{\sqrt{1+x^4}}{1-x^4}dx$
Have you tried $x^4=\tan^2u$ or $x^4=\sinh^2v$ ?
1d
comment Why is $\sum_{r=1}^{m-1} (2r+1)r=\sum_{r=1}^{m-1} 4\binom{r}{2} + 3\binom{r}{1}$?
See Faulhaber's formulas.
1d
answered who discovered the orthocenter of a triangle?
1d
revised Finding $ \int_0^1 \frac {\ln x}{1+x^2}\mathrm dx $
Removed % for Compatibility.
1d
comment An infinite series of integrals $\int_{0}^{\eta}\cos nt\log\left(\frac{\cos(t/2)+\sqrt{\cos^2(t/2) -\cos^2(\eta/2)}}{\cos(\eta/2)}\right) dt$
For this integral I know the most marvelous proof, which this comment section is too narrow to contain$\ldots$
1d
comment How find this sum closed form $I=\sum_{k=1}^{n}\int_{0}^{+\infty}\cos{(2kx)}x^{m-1}e^{-ax}dx$
@math110: This is due to the fact that $x^me^{ax}$ is noting else than the $m^{th}$ derivative of $e^{ax}$ with regard to a.
2d
comment How evaluate the following hard integrals?
$1-\cos2x=2\sin^2x$.
2d
awarded  Constituent
2d
revised How to compute $\sum_{n\ =\ 1}^{\infty}\arctan\left(\,\frac{3n^{2}}{ 2n^{4} - 1}\,\right)$
Removed % for Compatibility.
2d
comment Closed-form of infinite continued fraction involving factorials
See OEIS A$100608$.
Dec
19
comment When can $n^k+k$ be a perfect square?
@Peter: Yes, of course. $($This has already been mentioned in the OP$)$.
Dec
19
answered Closed form for $\sum_{t=0}^{n} t^2x^t$
Dec
19
answered How to integrate a fraction of the type $\frac{1}{(ax+b)^c(dx+e)^f}$?
Dec
19
comment Sum of all n dimensional spheres?
The sum of all n-dimensional hyper-cubes: $$\sum_{n=0}^\infty x^n~=~\dfrac1{1-x}$$ The sum of all n-dimensional straight-edged pyramids: $$\sum_{n=0}^\infty\frac{x^n}{n!}~=~e^x$$