# An upper bound of an integral

For any $$a, i want to prove that there exists $$M_0 \in \mathbb{R}$$ such that \begin{align} \left| \int_a^b \frac{\sin x}{x} \text{ d}x \right| \leq M_0. \end{align} Here, $$M_0$$ is independent of $$a$$ and $$b$$. How can i prove this? i can find an upper bound which (unfortunately) depends on $$a$$ and $$b$$.

I appreciate if anyone gives me a hint.

Thanks.

• Integration by parts – mathworker21 Mar 28 at 19:43
• More hint: specifically, the integration by parts creates a sufficient decay in the denominator for you to have a boundary – UnbelieveTable Mar 28 at 19:49

The proof of this bound should be similar in character to the proof that the improper integral of $$\frac{\sin x}{x}$$ converges. Over large intervals, the integral behaves like an alternating series, with positive and negative bumps that partially offset. I see two ways to go with this:
• Leverage the alternating series estimate. If we're working with full bumps, so that $$a$$ and $$b$$ are integer multiples of $$\pi$$, the bumps alternate in sign and decrease in value as we go to $$\infty$$. The alternating series estimate tells us that the sum of these is no more than the first term in absolute value.
• Integration by parts. We're not trying to find an antiderivative; we're instead trying to convert this into an absolutely convergent integral, and use this to get an estimate. By the way, the appropriate antiderivative for $$\sin x$$ is $$1-\cos x$$. We need this to not blow up at zero.
These are not complete arguments, of course; they're merely enough to get started. I'll also note that the best possible $$M_0$$ here is $$\int_{-\pi}^{\pi} \frac{\sin x}{x}\,dx$$.
• I get a maximum integral of about $3.704$, achieved with bounds $b=\pi$ and $a=-\pi$. – jmerry Mar 29 at 8:19
• Yes i saw that too. I plotted the function and verified that the largest possible area is the one between the curve, the $x$-axis and the vertical lines $x=-\pi$ and $x=\pi$. But how can i prove this algebraically? – Hussein Eid Mar 29 at 8:26