A bounded integral I want to show that there exists $K\in\mathbb{R}^+$ such that
$$\left|\int_{1}^x \sin(t+t^7)dt \right|<K$$
for all $x\ge 1$. Intuitively, I'm quite sure it is true, but I can't find a formal proof. Any idea?

Let $f(x) = x + x^7$, and $g(t)$ its inverse function on $[0,\infty)$.
Then $$\int_0^x \sin(x + x^7)\ dx = \int_0^{f(x)} \sin(t) g'(t)\ dt$$
It can be shown that as $t \to \infty$, 
$$g(t) = t^{1/7} - \dfrac{1}{7} t^{-5/7} + O(t^{-11/7})$$
and 
$$g'(t) = \dfrac{1}{7 g(t)^6 + 1} =  \dfrac{1}{7} t^{-6/7} + O(t^{-12/7})$$
Now $\int_1^\infty |\sin(t)| t^{-12/7}\ dt < \infty$.
On the other hand, using Integration by Parts,
$$ \int_1^R \sin(t) t^{-6/7}\ dt = \left. - \cos(t) t^{-6/7} \right|_1^R - 
\dfrac{6}{7} \int_1^R \cos(t) t^{-13/7}\ dt$$ 
and again the term on the right is bounded.
