So I am trying to solve the following integral and apparently its not integrable or I might be wrong. Not even computer software can integrate.
Can anyone tell me if this is integrable or not? The function is the following,
$$\int_{y}^{+\infty} \left(\frac{ \left(1-\exp \left(- x^{0.25}\right)\right)}{x^{0.5}}+\exp \left(- \sqrt{x}\right)\right) \left(\exp \left(-s \ x^{-1}\right)-1\right) \mathrm dx$$
where $s$ is just a constant and $y$ is a positive constant.
Hint:
$\int_y^\infty\left(\dfrac{1-e^{-\sqrt[4]x}}{\sqrt{x}}+e^{-\sqrt{x}}\right)(e^{-\frac{s}{x}}-1)~dx$
$=\lim\limits_{z\to\infty}\int_y^z\left(\dfrac{1-e^{-\sqrt[4]x}}{\sqrt{x}}+e^{-\sqrt{x}}\right)(e^{-\frac{s}{x}}-1)~dx$
$=\lim\limits_{z\to\infty}\left(\int_y^z\dfrac{(1-e^{-\sqrt[4]x})(e^{-\frac{s}{x}}-1)}{\sqrt{x}}dx+\int_y^ze^{-\sqrt{x}}(e^{-\frac{s}{x}}-1)~dx\right)$
$=\lim\limits_{z\to\infty}\left(\int_y^z2(1-e^{-\sqrt[4]x})(e^{-\frac{s}{x}}-1)~d(\sqrt{x})+\int_y^ze^{-\sqrt{x}}(e^{-\frac{s}{x}}-1)~dx\right)$
$=\lim\limits_{z\to\infty}\left(\int_\sqrt{y}^\sqrt{z}2(1-e^{-\sqrt{x}})(e^{-\frac{s}{x^2}}-1)~dx+\int_y^ze^{-\sqrt{x}}(e^{-\frac{s}{x}}-1)~dx\right)$