Calculate a certain integral Good day! Is this integral tabular? I calculated it in MatLab and am now trying to write down an analytical expression. How can I get a result?
\begin{align}
I &= \int\limits_{x=-\infty}^{-1} \frac{\mu \, dx}{2 \cdot (1+\mu^2 \cdot ((x-m) \cdot k)^2)^{3/2}} \\ 
&= \left[- \frac{\mu \cdot (m-x)}{2 k \mu \sqrt{(m-x)^2+1}} \right]_{x = -\infty}^{-1} \\
&= - \frac{\mu \cdot (m+1)}{2 k \mu \sqrt{(m+1)^2+1}} + 
\lim_{x\to-\infty} \, \frac{\mu \cdot (m-x)}{2 k \mu \sqrt{(m-x)^2+1}} \\
&= - \frac{\mu \cdot (m+1)}{2 k \mu \sqrt{(m+1)^2+1}} + 
\frac{\mu \cdot (m+\infty)}{2 k \mu \sqrt{(m+\infty)^2+1}} \\
&= \frac{1}{2 k} - \frac{\mu \cdot (m+1)}{2 k \mu \sqrt{(m+1)^2+1}}.
\end{align}

The integral
$$ I = \int\limits_{x=-\infty}^{-1} \frac{\mu \, dx}{2 \cdot (1+\mu^2 \, k^2 \, (x-m)^2)^{3/2}} $$
can be evaluated as follows.
\begin{align}
I &= \int\limits_{x=-\infty}^{-1} \frac{\mu \, dx}{2 \cdot (1+\mu^2 \cdot ((x-m) \cdot k)^2)^{3/2}} \\
&= \frac{\mu}{2} \, \int_{1}^{\infty} \frac{dt}{(1 + \mu^2 \, k^2 \, (m+t)^2)^{3/2}} \hspace{5mm} \to x = -t \\
&= \frac{\mu}{2} \, \left[ \frac{m+t}{\sqrt{\mu^2 \, k^2 \, (m+t)^2 + 1}} \right]_{1}^{\infty} \\
&= \frac{\mu}{2} \, \left[ \lim_{t \to \infty} \frac{1}{\sqrt{\mu^2 \, k^2 + \frac{1}{(m+t)^2}}} - \frac{m+1}{\sqrt{\mu^2 \, k^2 \, (m+1)^2 + 1}} \right] \\
&= \frac{\mu}{2} \, \left[ \frac{1}{\mu \, k} - \frac{m+1}{\sqrt{\mu^2 \, k^2 \, (m+1)^2 + 1}} \right] \\
&= \frac{1}{2 \, k} - \frac{\mu \, (m+1)}{2 \, \sqrt{\mu^2 \, k^2 \, (m+1)^2 + 1}}
\end{align}
