Integration $\int_{0}^{\infty}\frac{b^2 \sin(ax)}{x^2+b^2}\,dx$

Question

$$\int_{0}^{\infty}\frac{b^2 \sin(ax)}{x^2+b^2}\,dx$$

Integration of such function using inverse Laplace transform is possible or other approaches should be applied?

Answer 1

First of all we can manage the integral a bit, rewriting it as ($ax = y$)

$$ab^2\int_0^{+\infty} \frac{\sin(y)}{y^2 + c^2}\ \text{d}y$$

Where $c = ab$.

Again, we can transform it easily as

$$ab^2\left(\frac{1}{c}\int_0^{+\infty} \frac{\sin(cs)}{1 + s^2}\ \text{d}s\right)$$

And again ($c = ab$)

$$b\int_0^{+\infty} \frac{\sin(abs)}{1 + s^2}\ \text{d}s$$

And it depends on special functions such as Cosine-Integral and the Hyperbolic-Sine-Integral. The result of the integral is this:

$$-\left(\text{Ci}\left(-i ab\right) + \text{Ci}\left(i ab\right)\right) \sinh \left(ab\right)+ 2 \text{Shi}\left(ab\right) \cosh \left(ab\right)$$

Where

$Ci$ is the CosIntegral special function;

$Shi$ is the Hyperbolic Sine Interal

More here: https://en.wikipedia.org/wiki/Trigonometric_integral#Cosine_integral and here: https://en.wikipedia.org/wiki/Trigonometric_integral#Hyperbolic_sine_integral

Consider that, as $(a, b)> 0 we can use Meyer G Function:

$$\frac{1}{4} \sqrt{\pi } a b G_{1,3}^{2,1}\left(\frac{a^2 b^2}{4}| \begin{array}{c} 0 \\ 0,0,-\frac{1}{2} \\ \end{array} \right)$$

More here: https://en.wikipedia.org/wiki/Meijer_G-function