If range of function `f(x)=sin^(-1)x+2tan^(-1)x+x^2+4x+1` is `[p , q],` then the value of `(p+q)` is_______>

Explanation

To find the value of \( p + q \) where the range of the function \( f(x) = \sin^{-1}(x) + 2\tan^{-1}(x) + x^2 + 4x + 1 \) is given as \([p, q]\), we will follow these steps:

### Step 1: Determine the Domain of the Function
The function \( f(x) \) contains \( \sin^{-1}(x) \) which is defined for \( x \in [-1, 1] \). The function \( \tan^{-1}(x) \) is defined for all real numbers. Therefore, the domain of \( f(x) \) is:
\[ x \in [-1, 1] \]

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