If \( f(x) \) and \( g(x) \) are functions such that \( f(x)=\displaystyle\frac{d}{dx}\left(2025-\sqrt{x}\right) \) and \( g(x)=\displaystyle\frac{d}{dx}(x+2025) \), then
\( \int\left[f(x)+g(x)\right]dx \) is equal to
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( 4050-\sqrt{x}+x+C \) : where \( C \) is an arbitrary constant
Since f and g are themselves derivatives, f + g is the derivative of
\( \displaystyle (2025-\sqrt{x})+(x+2025)=4050-\sqrt{x}+x \)
Integration undoes differentiation, so integrating f + g gives this function back, up to a constant:
\( \displaystyle \int\left[f(x)+g(x)\right]dx=4050-\sqrt{x}+x+C \)