∫(1-u^2/c^2)^(-3/2)du
You asked:
Evaluate the integral: \(\int{{\left( 1 - \frac{{u}^{2}}{{c}^{2}} \right)}^{\frac{-3}{2}}}du\)
MathBot Answer:
Antiderivative
\[\int{{\left( 1 - \frac{{u}^{2}}{{c}^{2}} \right)}^{\frac{-3}{2}}}du = C + \begin{cases} - \frac{i u}{\sqrt{-1 + \frac{u^{2}}{c^{2}}}} & \text{for}\: \left|{\frac{u^{2}}{c^{2}}}\right| > 1 \\\frac{u}{\sqrt{1 - \frac{u^{2}}{c^{2}}}} & \text{otherwise} \end{cases}\]
\(i\) is the imaginary unit, defined as \(i^2 = -1\).