Definite integral of $\sin(x)/x$ I'm trying to evaluate absolute value of:
$$\left | \int_{a}^{b} \frac{\sin(x)}{x} dx \right | \leq\frac{2}{a}, \ \text{where}\ 0<a<b$$
I tried taylor series, mean value theorems, and tried to divide interval on some like $\left[\pi k, \pi(k+1) \right]$ or $\left[2\pi k, 2\pi(k+1) \right]$, but it didn't help

Use
$$ \int\frac{\sin x}xdx=-\int\frac1xd\cos x=-\frac{\cos x}{x}-\int\frac{\cos x}{x^2}dx.$$
one has
\begin{eqnarray}
&&\bigg|\int_a^b\frac{\sin x}xdx\bigg|\\
&=&\bigg|-\frac{\cos x}{x}\bigg|_a^b-\int_a^b\frac{\cos x}{x^2}dx\bigg|\\
&\le&\bigg|\frac{\cos a}{a}-\frac{\cos b}{b}\bigg|+\int_a^b\frac1{x^2}dx\\
&\le&\frac{1}{a}+\frac1b+\frac1a-\frac1b\\
&=&\frac{2}{a}.
\end{eqnarray}
