How can I study the convergence of the improper integral $\int_{0}^{ \infty} \frac{\sin(x)}{x+1} \, \mathrm dx\,$?

Solution 1:

Notice that $$\int_0^\infty \frac{\sin x}{x+1}\,dx = \frac{-\cos x}{x+1}\Bigg|_0^\infty - \int_0^\infty \frac{\cos x}{(x+1)^2}\,dx = 1 - \int_0^\infty \frac{\cos x}{(x+1)^2}\,dx$$

and the last integral converges absolutely since $$\int_0^\infty \frac{\left|\cos x\right|}{(x+1)^2}\,dx \le \int_0^\infty \frac{dx}{(x+1)^2} = \int_1^\infty \frac{dx}{x^2} < +\infty.$$

However the original integral does not converge absolutely. Namely, we have $$x \in \bigcup_{k \in \mathbb{N}_0} \left[\frac\pi6+k\pi,\frac{5\pi}6+k\pi\right] \implies \left|\sin x\right| \ge \frac12$$ so $$\int_0^\infty \frac{\left|\sin x\right|}{x+1}\,dx \ge \frac12\sum_{k=0}^\infty \int_{\frac\pi6+k\pi}^{\frac{5\pi}6+k\pi} \frac{dx}{x+1} = \frac12\sum_{k=0}^\infty \ln \frac{\frac{5\pi}6+k\pi+1}{\frac\pi6+k\pi+1} = +\infty.$$

Solution 2:

The Cauchy criterion for improper integrals is:

An improper integral $\int_0^\infty f(x) \, dx$ is convergent if and only if for any $\epsilon > 0$ there exists $C_\epsilon > 0$ such that $\left|\int_a^b f(x) \, dx \right| < \epsilon$ for all $b > a> C_\epsilon.$

Since $x \mapsto \frac{1}{1+x}$is decreasing, by the second mean value theorem for integrals, there exists $\xi \in (a,b)$ such that

$$\left|\int_a^b \frac{\sin x}{1+x} \, dx\right| = \left|\frac{1}{1+a}\int_a^\xi \sin x\, dx\right| = \frac{|\cos a - \cos \xi|}{1+a}\leqslant \frac{2}{1+a}$$

For all $b > a > C_\epsilon = \frac{2}{\epsilon}-1$ we have the RHS less than $\epsilon$ and the Cauchy criterion is satisfied.

Solution 3:

Granted, the integral does not converge in the sense of Lebesgue. As a proper Riemann integral it does.

Here is another solution based which uses elementary facts about alternating series.

• The sequence $a_n=\Big|\int^{(n+1)\pi}_{n\pi}\frac{\sin x}{x+1}\,dx\Big|$ is non decreasing and $a_n\xrightarrow{n\rightarrow\infty}0$. This is because on $[\pi n,\pi(n+1)]$, $\sin x=(-1)^n|\sin x|$, and so $$ \begin{align} a_{n+1}&=\int^{(n+2)\pi}_{(n+1)\pi}\frac{|\sin x|}{x+1}\,dx=\int^{(n+1)\pi}_{n\pi}\frac{|\sin(x+\pi)|}{x+\pi+1}\,dx\\ &\leq \int^{(n+1)\pi}_{n\pi}\frac{|\sin x|}{x+1}=a_n\leq\frac{\pi}{\pi n +1}\xrightarrow{n\rightarrow\infty}0 \end{align}$$

• The series $s=\sum_{n\geq0}(-1)^na_n$ has partial sums $s_n=\int^{n\pi}_0\frac{\sin x}{1+x}\,dx$. Being a nice alternating series, $s_n$ converges.

• In general, for $T>0$, let $[T]$ be its integer part. Then

$$ \Big|\int^{T\pi}_0\frac{\sin x}{x+1}\,dx - \int^{[T]\pi}_0\frac{\sin x}{x+1}\,dx\Big|\leq \int^{\pi T}_{[T]\pi}\frac{|\sin x|}{x+1}\leq \frac{\pi}{[T]\pi+1}\xrightarrow{T\rightarrow\infty}0$$

Therefore $\lim_{A\rightarrow\infty}\int^{A}_0\frac{\sin x}{x+1}\,dx$ exists and equal $s$.