How to integrate sinc function numerically

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How to compute (numerically)

$$ F(x) = \int_{-\infty}^x \dfrac{\sin(t)}{t} dt $$

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$$F(x) = \int_{-\infty}^x \dfrac{\sin(t)}{t} dt=\text{Si}(x)+\frac{\pi }{2}$$

For the computation of the sine integral function, you will find subroutines in Numerical Recipes (have a look here).