# Cauchy principle value of $\int_{-\infty}^{\infty}\sin(x)/(x-a)dx$

I need to find the cauchy principle value of $\int_{-\infty}^{\infty}\sin(x)/(x-a)dx$ ? I'm think of rewriting in terms of $e^{i\theta}$ and try to rewrite as contour integral?

Need some aid on how to calculate this integral please. Thank you for help

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All that is needed to translate the integrand to obtain

$$\mathrm{PV}\int_{-\infty}^{\infty}\frac{\sin(x+a)}{x}\,dx = \mathrm{PV}\int_{-\infty}^{\infty}\frac{\sin x \cos a + \cos x \sin a }{x}\,dx.$$

$$\int_{-\infty}^{\infty}\frac{\sin x}{x}\,dx = \pi \qquad\text{and}\qquad \mathrm{PV}\int_{-\infty}^{\infty}\frac{\cos x}{x}\,dx = 0.$$
$$\int_{-\infty}^{\infty}\frac{\sin x}{x}\,dx = \pi.$$