A function \(f\) is defined as \(f(x) = \sin x + 2x\).

Evaluate \(\large \displaystyle \int _{2+\sin \left(1\right)}^{4+\sin \left(2\right)}f^{-1}(x) \ dx\) correct to 3 decimal places.

**Clarifications:** \(f^{-1}(x)\) denotes the inverse of \(f(x)\), i.e. if \(y = f(x)\), then \(x = f^{-1}(y)\).

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