On Integral consisting $\sin^2(f(x))$:

Question

Consider the following integral:

$$I(t)=\int_1^t\sin^2(f(x))dx$$

Here , $f(x)$ is monotonic for the given domain and is at least twice differentiable.

Is there a result (in its full generality) of following type exists :

$I(x)= g(x)+O(j(x))$

Such that , $j(x)$ is bounded and g(x) is monotonic .