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$\begin{array}{cc}{\int }_{0}^{\infty }\left[\text{s}\text{i}\left(ax\right)+\frac{\pi }{2}\right]cosbx\frac{x\mathrm{d}x}{{x}^{2}+{c}^{2}}=-\frac{\pi }{4c}\left\{{e}^{-bc}\left[\text{E}\text{i}\left(bc\right)-\text{E}\text{i}\left(-ac\right)\right]+{e}^{bc}\left[\text{E}\text{i}\left(-bc\right)-\text{E}\text{i}\left(-ac\right)\right]\right\}\text{?}\text{?}\left[00\right]=\frac{\pi }{4c}{e}^{-bc}\left[\text{E}\text{i}\left(-ac\right)-\text{E}\text{i}\left(ac\right)\right]& \left[00\right]\end{array}$

BI (460)(2,5)

6.259

1.

$\begin{array}{ll}{\int }_{0}^{\infty }\text{s}\text{i}\left(ax\right)\text{?}\text{?}sinbx\frac{\mathrm{d}x}{{x}^{2}+{c}^{2}}& =\frac{\pi }{2c}\text{E}\text{i}\left(-ac\right)sinh\left(bc\right)\\ =\frac{\pi }{4c}\text{?}{e}^{-bc}\left[\text{E}\text{i}\left(-bc\right)+\text{E}\text{i}\left(bc\right)\text{?}\text{?}-\text{E}\text{i}\left(-ac\right)-\text{E}\text{i}\left(-ac\right)\text{?}\right]& \left[00\right]\\ +\frac{\pi }{2c}\text{E}\text{i}\left(-bc\right)sinh\left(bc\right)& \left[00\right]\end{array}$

ET I 96(8)

2.

$\begin{array}{lll}{\int }_{0}^{\infty }\text{c}\text{i}\left(ax\right)\text{?}\text{?}sinbx\frac{x\mathrm{d}x}{{x}^{2}+{c}^{2}}& =-\frac{\pi }{2}sinh\left(bc\right)\text{E}\text{i}\left(-ac\right)& \left[00\right]\\ =-\frac{\pi }{2}sinh\left(bc\right)\text{E}\text{i}\left(-bc\right)+\frac{\pi }{4}{e}^{-bc}\left[\text{E}\text{i}\left(-bc\right)+\text{E}\text{i}\left(bc\right)\\ -\text{E}\text{i}\left(-ac\right)-\text{E}\text{i}\left(-ac\right)\right]& \left[00\right]\end{array}$

BI (460)(3)a, ...

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