Integral $\int e^{-\theta^{\beta}}(\theta^{\beta})^{n}d\beta$

Question

For parameter $\theta$, i was integrating $\int e^{-\theta^{\beta}}(\theta^{\beta})^{n}d\beta$, because i can't solve it using substitusion or partial integral, so i used wolframalpha to calculate it, but the outcome is, (no result found in terms of standard mathematical function). What 'not standard' mathematical function that could solve the integral? And if i give lower and upper bound, is it now be able to calculated? may be by numerically or something?

Answer 1

With CAS Help and Mellin Transform:

$$\color{blue}{\int e^{-\theta ^{\beta }} \theta ^{n \beta } \, d\beta} =\\\int \mathcal{M}_A\left[e^{-A* \theta ^{\beta }} \theta ^{n \beta }\right](s) \, d\beta =\\\int \theta ^{n \beta } \left(\theta ^{\beta }\right)^{-s} \Gamma (s) \, d\beta =\\\mathcal{M}_s^{-1}\left[\frac{\theta ^{n \beta } \left(\theta ^{\beta }\right)^{-s} \Gamma (s)}{n \log (\theta )-s \log (\theta )}\right](1)=\\-\frac{\pi \csc (n \pi ) \left(\Gamma (1+n)-n \Gamma \left(n,0,\theta ^{\beta }\right)\right)}{\Gamma (1-n) \Gamma (1+n) \log (\theta )}+C=\color{blue}{\\-\frac{\Gamma \left(n,\theta ^{\beta }\right)}{\log (\theta )}+C}$$

where: $\Gamma \left(n,\theta ^{\beta }\right)$ is incomplete gamma function.