### 3.926 Trigonometric functions of more complicated arguments combined withexponentials

3.9216

1.12

${\int }_{0}^{\infty }{e}^{-\gamma \infty }cosa{x}^{2}\left(cos\gamma x-sin\gamma x\right)dx=\sqrt{\frac{\pi }{8a}}\mathrm{exp}\left(-\frac{{\gamma }^{2}}{2a}\right)$ ET | 26(28)

[a > 0, Re γ ≥ |Im γ|]

2.10

${\int }_{0}^{\pi /4}\underset{n=1}{\overset{\infty }{\Pi }}\mathrm{exp}\left[-\frac{1}{n}{tan}^{2n}x\right]=\frac{\pi }{2}-1$ 3.10

${\int }_{0}^{\pi /2}\mathrm{exp}\left[-\sum _{n=1}^{\infty }\frac{1}{n}{sin}^{2n}x\right]={\int }_{0}^{\pi /2}\mathrm{exp}\left[-\sum _{n=1}^{\infty }\frac{1}{n}{cos}^{2n}x\right]=\frac{\pi }{4}$ 3.922

1.

$\begin{array}{ll}{\int }_{0}^{\infty }{e}^{-\beta {x}^{2}}sina{x}^{2}\text{d}x=\frac{1}{2}{\int }_{-\infty }^{\infty }{e}^{-\beta {x}^{2}}sina{x}^{2}\text{d}x\hfill & =\sqrt{\frac{\pi }{8}}\sqrt{\frac{\sqrt{{\beta }^{2}+{a}^{2}}-\beta }{{\beta }^{2}+{a}^{2}}}\hfill \\ =\frac{\sqrt{\pi }}{2\sqrt{{\beta }^{2}+{a}^{2}}}sin\left(\frac{1}{2}\mathrm{arctan}\frac{a}{\beta }\right)\hfill \end{array}$

FI II 750, BI (263)(8)

[Reβ > 0, a>0]

2.

$\begin{array}{ll}{\int }_{0}^{\infty }{e}^{-\beta {x}^{2}}cosa{x}^{2}\text{d}x=\frac{1}{2}{\int }_{-\infty }^{\infty }{e}^{-\beta {x}^{2}}cosa{x}^{2}\text{d}x\hfill & =\hfill \end{array}$

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