\(\int_0^\infty\frac{|\sin{x}|+|\cos{x}|}{x+1}dx\)

[Grigorios Kostakos]

Examine if the integral \(\displaystyle\int_{0}^{+\infty}{\frac{|{\sin{x}}|+|{\cos{x}}|}{x+1}dx}\) converges or not.

[Papapetros Vaggelis]

The answer is negative.

For all \(\displaystyle{x\geq 0}\), we have

\(\displaystyle{0\leq |{\sin x}|\leq 1\Rightarrow |{\sin x}|\geq \sin^2 x}\)

\(\displaystyle{0\leq |{\cos x}|\leq 1\Rightarrow |{\cos x}|\geq \cos^2 x}\)

So, for all \(\displaystyle{x\geq 0}\), we have

\(\displaystyle{|{\sin x}|+|{\cos x}|\geq \sin^2 x+\cos^2 x=1\Rightarrow \frac{|{\sin x}|+|{\cos x}|}{x+1}\geq \frac{1}{x+1}}\)

and

\(\displaystyle{\int_{0}^{+\infty}\frac{1}{x+1}\,dx=\lim_{x\to +\infty}\int_{0}^{x}\frac{1}{t+1}\,dt=\lim_{x\to +\infty}\ln (1+x)=+\infty}\)

Therefore, \(\displaystyle{\int_{0}^{+\infty}\frac{|{\sin x}|+|{\cos x}|}{x+1}\,dx=+\infty}\)