I have to analyze if the following function is Lebesgue integrable: $\space$
$f(x)=\frac{\cos x}{\sqrt{x+1}}$ $\space$, $x\in(0,+\infty)$
Sorry but I have no idea how to do this exercise... I tried to find information about it but I don't even know how to start with it... Could someone help me?
Hint/comments to get started: A function $f$ is integrable on a set $E$ if $\int_E|f|$ is finite. For $x$ near $0$, $f(x)$ looks like the constant function $1$, so you can focus on $f(x)$ for values of $x$ larger than $1$, say. So, is $\int_{1}^\infty|f|$ finite? Well, if we ignore the $\cos(x)$ term, then $|f(x)| = 1/\sqrt{x+1}$, which is basically $1/\sqrt{x}$ for $x$ larger than $1$. You should recognize that $f(x) = 1/\sqrt{x}$ is not integrable since its definite integral $\int_1^xf(t)\,dt \approx \sqrt{x}$, which tends to $\infty$ as $x$ gets larger.
So, what about when we include $\cos(x)$? What effect does that have on the function? Well, on intervals where $\cos(x)\ge 1/2$, the function we have is $\ge 1/2\sqrt{x+1}$. First, what are those intervals? Second, what can you say about the relationship between $\int |f|$ and the integral of $|f|$ on just those intervals we described? Third, what can you conclude about the integrability of $f$?
To make your intuition into a fully rigorous proof, you may need to use a convergence theorem such as the monotone convergence theorem.