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

3.926 Trigonometric functions of more complicated arguments combined withexponentials

3.921⁶

1.¹² 

${\displaystyle {\int }_0^{\infty }{e}^{-\gamma \infty }cosa{x}^2\left(cos\gamma x-sin\gamma x\right)dx=\sqrt{\frac{\pi }{8a}}\exp \left(-\frac{{\gamma }^2}{2a}\right)}$

  ET | 26(28)

[a > 0, Re γ ≥ |Im γ|]

2.¹⁰ 

${\displaystyle {\int }_0^{\pi /4}\underset{n=1}{\overset{\infty }{\Pi }}\exp \left[-\frac{1}{n}{tan}^{2n}x\right]=\frac{\pi }{2}-1}$

3.¹⁰ 

${\displaystyle {\int }_0^{\pi /2}\exp \left[-{\displaystyle \sum _{n=1}^{\infty }\frac{1}{n}{sin}^{2n}x}\right]={\displaystyle {\int }_0^{\pi /2}\exp \left[-{\displaystyle \sum _{n=1}^{\infty }\frac{1}{n}{cos}^{2n}x}\right]}}=\frac{\pi }{4}$

3.922

1. 

$\begin{array}{ll}{\displaystyle {\int }_0^{\infty }{e}^{-\beta x^2}sina{x}^2\text{d}x=\frac{1}{2}{\displaystyle {\int }_{-\infty }^{\infty }{e}^{-\beta x^2}sina{x}^2}\text{d}x}\hfill & =\sqrt{\frac{\pi }{8}}\sqrt{\frac{\sqrt{{\beta }^2+a^2}-\beta }{{\beta }^2+a^2}}\hfill \\ =\frac{\sqrt{\pi }}{2\sqrt[4]{{\beta }^2+a^2}}sin\left(\frac{1}{2}\arctan \frac{a}{\beta }\right)\hfill \end{array}$

  FI II 750, BI (263)(8)

[Reβ > 0, a>0]

2. 

$\begin{array}{ll}{\displaystyle {\int }_0^{\infty }{e}^{-\beta x^2}cosa{x}^2\text{d}x=\frac{1}{2}{\displaystyle {\int }_{-\infty }^{\infty }{e}^{-\beta x^2}cosa{x}^2}\text{d}x}\hfill & =\hfill \end{array}$