I'm trying to calculate an integral of the form:
$$ \textrm{Int}=PP\int_{1}^{\infty} \dfrac{x ^2 a^2 + 2 a x+ 2}{\sqrt{x ^2-1} \left[\left(b^2-c^2\right)x ^2 -b^2\right]} e^{-ax}\, dx $$
where $a>0,\,b>0,\,c>0$ are reals and $PP$ is the Cauchy principale value of the integral.
please, is this true for my $ \textrm{Int} $,
$$ PP\int_{1}^{\infty} \dfrac{x ^2 a^2 + 2 a x+ 2}{\sqrt{x ^2-1} \left[\left(b^2-c^2\right)x ^2 -b^2\right]} e^{-ax}\, dx = \int_{1}^{\infty} \dfrac{x ^2 a^2 + 2 a x+ 2}{\sqrt{x ^2-1} \left[\left(b^2-c^2\right)x ^2 -b^2\right]} e^{-ax}\, dx $$
how to calculte it please ?
The problem is that the integrand may have a pole between $1$ and $\infty$, so that the integral itself diverges.
If $b>c$, then the denominator vanishes at $x=q$, where $$q=\frac{b}{\sqrt{b^2-c^2}}$$ So the "principal value" means do an integral $$ \int_1^{q-\epsilon} + \int_{q+\epsilon}^\infty $$ and take the limit as $\epsilon \to 0^+$.