91Ó°ÊÓ

Integration is the process of finding a function when its derivative is given. In this problem, we start with the third derivative of a function, which is expressed as: \[ f^{\frac{1}{1}}(x) = \frac{1}{\text{√}(1 + \text{sin}^2(x))} \] To retrieve the original function from this derivative, we need to integrate it three times. Each integration step reverses one derivative, gradually taking us closer to the original function. - In the first step, we integrate the third derivative to get the second derivative, denoted as \( f^{\frac{1}{2}}(x) \). - Then, we integrate the second derivative to obtain the first derivative, \( f^{\frac{1}{1}}(x) \). - Finally, we integrate the first derivative to find the original function, \( f(x) \).Remember, every time we integrate, we add a constant of integration, since the derivative of any constant is zero.

The third derivative of a function indicates how the second derivative changes with respect to \( x \). In other words, it tells us about the concavity and the rate at which the curvature of the function is changing. Given the third derivative \( f^{\frac{1}{1}}(x) = \frac{1}{\text{√}(1 + \text{sin}^2(x))} \), we use integration to find the lower-order derivatives and ultimately the original function. The process follows these steps:

• First, integrate the third derivative to find the second derivative \( f^{\frac{1}{2}}(x) \).
• Second, integrate the second derivative to find the first derivative \( f^{\frac{1}{1}}(x) \).
• Third, integrate the first derivative to find the original function \( f(x) \).

Note that each integration step involves adding an integration constant to account for any lost constants from differentiation.

In calculus, the constant of integration appears whenever we integrate a function. When differentiating, any constant term disappears, so integrating requires adding back an unknown constant to account for all possible original functions. For example, after integrating the third derivative \( \frac{1}{\text{√}(1 + \text{sin}^2(x))} \), we get: \[ f^{\frac{1}{2}}(x) = \text{∫} \frac{1}{\text{√}(1 + \text{sin}^2(x))} \text{dx} + C_1 \] Here, \( C_1 \) is the constant of integration. - The same process repeats for each subsequent integration, leading to more integration constants \( C_2, C_3, \) and \( C_4 \). - Each constant represents a level of freedom in the possible solutions for the function. Ultimately, the general solution is expressed as a combination of several integration constants, reflecting the family of functions that share the same derivatives up to the third order.