TITLE: Explicit form for $I(n) = \int_{0}^{1} \frac {(1-x)^n}{1+x^2} dx$
QUESTION [1 upvotes]: Is it possible to find an explicit form for the Integral $$I(n) = \int_{0}^{1} \frac {(1-x)^n}{1+x^2} dx$$
for $n \ge 0$?
I tried differentiating under the Integral sign but it doesnt seem to help.
If we try to replace either $(1-x)^n$ or $\frac 1{1+x^2}$ with an infinite series representation, it gets messier. We can get that $I(n)$ is equal to $\sum_{k=0}^{\infty} (-1)^{k}\frac{n!(2k)!}{(2k+n+1)!}$ but I want a more explicit representation. 
If I plug the Integrals into Wolfram alpha, I can see that there is some kind of pattern involving $\pi$ and $\ln(2)$. That should be intuitiv, because the infinite sum above is similar to
$$\sum_{k=1}^{\infty} \frac {(-1)^k}{k} = -\ln(2)$$
and 
$$\sum_{k=1}^{\infty} \frac {(-1)^k}{2k-1} = -\frac{\pi}{4}$$
So my question is, is it possible to find a ecplicit form for $I(n)$, and does it involve $\pi$ and $\ln(2)$?

REPLY [1 votes]: Using geometric series and Euler's Beta function
$$ I(n) = \int_{0}^{1}(1-x)^n\sum_{k\geq 0}(-1)^k x^{2k}\,dx =\sum_{k\geq 0}(-1)^k \frac{\Gamma(n+1)\Gamma(2k+1)}{\Gamma(n+2k+2)}=\sum_{k\geq 0}(-1)^k\frac{n!(2k)!}{(2k+n+1)!}.$$
On the other hand, by partial fraction decomposition
$$ \frac{n!(2k)!}{(2k+n+1)!}=\frac{n!}{(2k+1)_{n+1}}=\sum_{h=0}^{n}\frac{(-1)^h \binom{n}{h}}{2k+h+1}$$
so
$$ I(n) = \sum_{k\geq 0}(-1)^k \sum_{h=0}^{n}\frac{(-1)^h \binom{n}{h}}{2k+h+1}=\sum_{h=0}^{n}(-1)^h\binom{n}{h}\sum_{k\geq 0}\frac{(-1)^k}{2k+h+1} $$
is a linear combination of the tails of the series
$$ \sum_{k\geq 0}\frac{(-1)^k}{2k+1}=\frac{\pi}{4}\qquad \sum_{k\geq 0}\frac{(-1)^k}{2k+2} = \frac{\log 2}{2}$$
as suspected.
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