Let \(f(x) =(1-x)^{2}\sin^2{x}+x^{2}\) for all \( x\in \mathbb{R}\), and let \(g(x)=\int_{1}^{x}\left(\frac{2(t-1)}{t+1}-\ln t\right)f(t)dt\) for all \(x\in(1,\infty)\).
Which of the following is true?
1
\(g\) is increasing on \((1, \infty)\)
2
\(g\) is decreasing on \((1, \infty)\)
3
\(g\) is increasing on (1, 2) and decreasing on \((2, \infty)\)
4
\(g\) is decreasing on (1, 2) and increasing on \((2, \infty)\)