Final Answer:
The integral ∫(x² - 25 - x²) dx simplifies to ∫(-25) dx, which equals -25x + c, where c is the constant of integration.
Step-by-step explanation:
The given integral ∫(x² - 25 - x²) dx involves the subtraction of x² from x², resulting in the constant term -25. Integrating a constant with respect to x yields -25x.
The integral of -25 is -25x, where x represents the variable of integration. Additionally, the constant of integration 'c' is added to account for any constant that might have been present in the original function but does not appear explicitly in the derivative.
Therefore, the solution to the integral is -25x + c, where 'c' denotes the constant of integration. This constant 'c' represents an arbitrary constant term that could have been present in the original function, as integrating a derivative results in multiple possible functions that differ only by a constant. Thus, the indefinite integral of the expression x² - 25 - x² simplifies to -25x + c, encapsulating the antiderivative of the given function.