Evaluate integral $\int\limits_1^\infty x^{y^m}dy$ What would be the evaluation of the definite integral $$\int\limits_1^\infty x^{y^m}dy$$ where $x$ with $0 < x < 1 $ is real ; $m > 1$ is any integer. At least an approximation will suffice if the error of the approximation is determinable.

Using the incomplete gamma function, this is $$ \frac{\Gamma(1/m,-\log x)}{m\cdot(-\log x)^{1/m}}. $$ Hint: Use the change of variable $t=-\log x\cdot y^m$.