If you meant $\displaystyle \int_{10}^{19}\frac{\sin(x)}{1+x^8}dx$:
$\displaystyle \int_{10}^{19}\frac{\sin(x)}{1+x^8}dx=-7.77628$ x$\displaystyle 10^-9$
This is calculated by wolfram alpha.
I dont think that this integral can be evaluated by using elementary mathematics.
If we take the indefinite integral $\displaystyle \int\frac{\sin(x)}{1+x^8}dx$:
This is also calculated wolfram alpha.