How Do You Compute This Improper Integral Involving Sine and a Limit?

[johnson12]

Hello and Happy New year, I'm having some trouble computing this integral:

lim_n->00\int^{1}_{0}\sqrt[3]{1+x^{n}sin(nx)}

Any suggestions are appreciated.

 

[Dick]

It sure looks like the limit ought to be one, doesn't it? Try to write down some inequalities sandwiching that integral between two integrals whose limit you know is one. Start by using -1<=sin(nx)<=1.