Evaluate limit of $\int_{x^2}^{x^3}t\sinh^{-1}(t^4)dt$

Question

I have the following question: find a finite non-zero limit or main part for the corresponding power function for the following integral:

$\lim_{x \to +\infty } \int_{x^2}^{x^3}t\sinh^{-1}(t^4)dt$

First I try to calculate this integral

$\int_{}^{}t\sinh^{-1}(t^4)dt$

After using Integration by Parts, I obtained:

$\frac{t^2}{2}\sinh^{-1}(t^4)-\int_{}^{}\frac{2t^5}{\sqrt{t^8+1}}dt$

However, I'm not sure how to deal with the right integral. I've searched in tables, but couldn't find a solution. Can this integral be solved using elementary functions?

Answer 1

There is no need to compute the integral. By taking the derivative, one can see that the integrand is increasing. In particular, $$ \int_{x^2}^{x^3} t\sinh^{-1}(t^4)\,dt\geq (x^3-x^2)\times x^2\sinh^{-1}(x^8)\to \infty $$ as $x\to \infty$.