较复杂的拉氏变换

青青子衿

\begin{align*}
\frac{\mathrm{d}}{\mathrm{d}x}I&=\frac{\mathrm{d}}{\mathrm{d}x}\int_{0}^{+\infty}\frac{\sin^{3}\left(at\right)\exp\left(-xt\right)}{t^{3}}\mathrm{d}t\\
&=-\int_{0}^{+\infty}\frac{\sin^{3}\left(at\right)\exp\left(-xt\right)}{t^{2}}\mathrm{d}t\\
&=\frac{3x}{4}\operatorname{arccot}\left(\frac{x}{a}\right)-\frac{x}{4}\operatorname{arccot}\left(\frac{x}{3a}\right)-\frac{3a}{8}\ln\left(\frac{9a^{2}+x^{2}}{a^{2}+x^{2}}\right)
\end{align*}

\begin{align*}
I&=\int_{0}^{+\infty}\frac{\sin\left(at\right)^{3}\exp\left(-xt\right)}{t^{3}}\mathrm{d}t=\,?
\end{align*}

\begin{align*}

\frac{3a^{2}\pi}{8}+\frac{3x^{2}}{8}\operatorname{arccot}\left(\frac{x}{a}\right)+\frac{3a^{2}}{8}\arctan\left(\frac{x}{a}\right)-\frac{x^{2}}{8}\operatorname{arccot}\left(\frac{x}{3a}\right)-\frac{9a^{2}}{8}\arctan\left(\frac{x}{3a}\right)-\frac{3ax}{8}\ln\left(\frac{9a^{2}+x^{2}}{a^{2}+x^{2}}\right)
\end{align*}


\begin{align*}
\color{black}{
\int_{0}^{\infty} \sin (nx) ( \cot x + \coth x ) e^{-nx} \ dx =
\left\{\begin{array}{cc}\frac{\pi}{2} \tanh (\frac{n \pi}{2} ),&\text{if n is odd}\\\frac{\pi}{2}\coth( \frac{n \pi}{2} ),&\text{if n is even}\end{array}\right.}
\end{align*}