Table of Integrals, Series, and Products, 8th Edition

$\begin{array}{ll}{\displaystyle {\int }_0^{\pi /2}cos(zcosx)cosax\text{ d}x}\hfill & =-asin\frac{a\pi }{2}{s}_{-1,}{a}_{}(z)\hfill \\ =-\frac{\pi }{4}\sec \frac{a\pi }{4}\left[{\text{J}}_a(z)+{\text{J}}_{-a}(z)\right]=\frac{\pi }{4}\text{cosec}\frac{a\pi }{2}\left[{\text{E}}_a(z)-{\text{E}}_{-a}(z)\right]\hfill \\ =-asin\frac{a\pi }{2}\left[-\frac{1}{a^2}+{\displaystyle \sum _{k=1}^{\infty }\frac{{(-1)}^{k-1}{z}^{2k}}{a^2\left(2^2-a^2\right)\left(4^2-a^2\right)\mathrm{\dots }\left[{(2k)}^2-a^2\right]}}\right]\hfill \end{array}$

  WA 339

[a > 0]

18. 

${\displaystyle {\int }_0^{\pi }cos(zcosx)cosnx\text{ d}x=\frac{1}{2}{\displaystyle {\int }_{-\pi }^{\pi }cos(zcosx)cosnx\text{ d}x=\pi cos\frac{n\pi }{2}}{J}_n(z)}$

  GW (334)(56b)

19. 

${\displaystyle {\int }_0^{\pi /2}cos(zcosx)cos2nx\text{ d}x}={(-1)}^n\frac{\pi }{2}{J}_{2n}(z)$

  WA 30(9)

20. 

${\displaystyle {\int }_0^{\pi /2}cos(zcosx){sin}^{2v}x\text{ d}x}=\frac{\sqrt{\pi }}{2}{\left(\frac{2}{z}\right)}^v\Gamma \left(v+\frac{1}{2}\right)J_v(z)$

  WA 35, WH

$\left[\mathrm{Re}v>-\frac{1}{2}\right]$

21. 

${\displaystyle {\int }_0^{\pi /2}cos(zcosx){sin}^{2\mu }x\text{ d}x}=\sqrt{\pi }{\left(\frac{2}{z}\right)}^{\mu }\Gamma \left(\mu +\frac{1}{2}\right)J_{\mu }(z)$

  WH

$\left[\mathrm{Re}\mu >-\frac{1}{2}\right]$