$\underset{0}{\overset{\infty }{\int }}x{\left({x}^{2}+{b}^{2}\right)}^{-\frac{1}{2}v}\text{?}\text{?}{\mathbit{K}}_{±v}\left(a\sqrt{{x}^{2}+{b}^{2}}\right)\text{?}sin\left(cx\right)\text{d}x=\sqrt{\frac{\pi }{2}{a}^{v}{b}^{v+\frac{3}{2}}c{\left({a}^{2}+{c}^{2}\right)}^{-\frac{1}{2}v-\frac{3}{4}}\text{?}{\mathbit{K}}_{-v-\frac{3}{2}}\left(b\sqrt{{a}^{2}-{c}^{2}}\right)}\text{?}$

ET I 113(45)

$?[Rea>0,??????Reb>0,?????c>0]$

411.

$\underset{0}{\overset{\infty }{\int }}x{\left({x}^{2}+{b}^{2}\right)}^{\mp \frac{1}{2}v}\text{?}\text{?}{\mathbit{K}}_{±v}\left(a\sqrt{{x}^{2}+{b}^{2}}\right)\text{?}cos\left(cx\right)\text{d}x=\sqrt{\frac{\pi }{2}{a}^{\mp v}{b}^{\frac{1}{2}v\mp }\text{?}{\left({a}^{2}+{c}^{2}\right)}^{±\frac{1}{2}v-\frac{1}{4}}\text{?}{\mathbit{K}}_{±v-\frac{1}{2}}\left(b\sqrt{{a}^{2}+{c}^{2}}\right)}\text{?}$

ET I 56(45)

$??[Rea>0,??????Reb>0,?????c??is?real]?$

5.

$\begin{array}{l}\underset{0}{\overset{\infty }{\int }}x{\left({x}^{2}+{a}^{2}\right)}^{-\frac{1}{2}v}\text{?}\text{?}{Y}_{v}\left(b\sqrt{{x}^{2}+{a}^{2}}\right)\text{?}cos\left(cx\right)\text{d}x=\sqrt{\frac{a\pi }{2}{\left(ab\right)}^{-v}\text{?}{\left({b}^{2}-{c}^{2}\right)}^{\frac{1}{2}v-\frac{1}{4}}\text{?}{Y}_{v-\frac{1}{2}}\text{?}\left(a\sqrt{{b}^{2}-{c}^{2}}\right)}\hfill \end{array}$

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