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Subsection 6.1.5 Some important Integrals:
\begin{equation*}
\int\limits_{0}^{\infty}\frac{\sin ax}{x}\,dx = \frac{\pi}{2}, \quad a \gt 0
\end{equation*}
\begin{equation*}
\int\limits_{0}^{\infty}e^{-ax} \sin bx \,dx = \frac{b}{a^{2}+b^{2}}
\end{equation*}
\begin{equation*}
\int\limits_{0}^{\infty}e^{-ax} \cos bx \,dx = \frac{a}{a^{2}+b^{2}}
\end{equation*}
\begin{equation*}
\int\limits_{0}^{\infty}e^{-x^{2}}\,dx = \frac{\sqrt{\pi}}{2}
\end{equation*}
\begin{equation*}
\int\limits_{-\infty}^{\infty}e^{-x^{2}}\,dx = 1
\end{equation*}
\begin{equation*}
\Gamma_{n}=a^{n}\int\limits_{0}^{\infty}e^{-ax}x^{n-1}\,dx, \quad \text{the Gamma function.}
\end{equation*}
