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

$\begin{array}{ll}{\displaystyle {\int }_0^{2\pi }\frac{{cos}^{2}x}{{\Delta }^3}dx}\hfill & =\frac{2{cos}^2\gamma }{p^2-q^2}\left[{sinh}^2\alpha {cosh}^2\beta \frac{I_1}{p^4}+{cosh}^2\alpha {sinh}^2\beta \frac{I_2}{q^4}\right]\hfill \\ +\frac{2{cosh}^2\alpha {cosh}^2\beta {sin}^2\gamma }{p^2{q}^2}\left(\frac{I_1}{p^2}-\frac{I_2}{q^2}\right)\hfill \end{array}$

7. 

$\begin{array}{ll}{\displaystyle {\int }_0^{2\pi }\frac{cosxsinx}{{\Delta }^3}dx}\hfill & =sinh\alpha cosh\alpha sinh\beta cosh\beta sin\gamma cos\gamma \hfill \\ \times \left[\frac{2}{p^2-q^2}\left(\frac{I_1}{p^4}+\frac{I_2}{q^4}\right)-\frac{2}{p^2{q}^2}\left(\frac{I_1}{p^2}-\frac{I_2}{q^2}\right)\right]\hfill \end{array}$

3.679(3)*

1. 

${\displaystyle {\int }_0^{\pi /2}\frac{dx}{\sqrt{a{cos}^{2}x+b{sin}^{2}x}}}=R_F\left(0,a,b\right)$

  DLMF (19.23.1)

[Rea > 0, Reb > 0]

2. 

${\displaystyle {\int }_0^{\pi /2}\sqrt{a{cos}^{2}x+b{sin}^{2}x}\text{d}x=2R_G\left(0,a,b\right)}$

  DLMF (19.23.2)

[Rea > 0, Reb > 0]

3. 

${\displaystyle {\int }_0^{}}$