Evaluate the following integral $$\int_0^\infty\frac{x+1}{x+2}\cdot\frac{x+3}{x+4}\cdot\frac{x+5}{x+6}\cdots dx$$
When I saw this, I was pretty sure that the infinite term must telescope or it must be a standard Progression or anything like that. But when I sat down to solve it, no simplifications were appearing. I also tried to isolate the $x+1$ and laboured to form a pattern but in vain. I wonder how to solve it.
Any help is greatly appreciated.
On
For each $x$, the integrand is a product of number strictly smaller than $1$ (and positive).
The integrand noted
$$f(x) := \prod_n u_n $$
Is a product of term strictly increasing and strictly smaller than $1$.
$$u_0(x) < u_n(x) <1, ~\forall n, x. $$
Then $$f(x) \leq \prod_i u_0(x), \forall x. $$
And this product is $0$ on real numbers since $u_0<1$.
We have $$ \prod_{k=1}^\infty \frac{x+(2k-1)}{x+2k} =\prod_{k=1}^\infty \left(1-\frac{1}{x+2k}\right)=0 $$ since for $0\le x\le 1$ $$ \sum_{k=1}^\infty \frac{1}{x+2k}\ge \sum_{k=1}^\infty \frac{1}{2+2k}=2\sum_{k=1}^\infty \frac{1}{1+k}= 2\left(\frac12+\frac13+\frac14+\dots\right) =\infty. $$ So the answer is $0$.