Find the find the function f given that the slope of the...

Find the find the function $f$ given that the slope of the tangent line to the graph of $f$ at any point $(x, f(x))$ is $f^{\prime}(x)$ and that the graph of $f$ passes through the given point.
$$f^{\prime}(x)=1-\frac{2 x}{x^{2}+1} ;(0,2)$$

Key Concepts

Integration Techniques for Rational Functions

Many functions, especially rational functions, require specialized integration techniques such as substitution, partial fractions, or algebraic manipulation to simplify the integrals into forms that are easier to evaluate. These techniques are essential for breaking down complex derivatives into simpler components that can be integrated individually.

Differential Equations and Initial Value Problems

This concept involves finding a function that satisfies a differential equation along with a given initial condition. It typically requires determining an antiderivative for the derivative provided and then using the initial condition to solve for the constant of integration. This approach is central to many problems in calculus and applied mathematics where a unique solution is necessary.

Antiderivatives and Indefinite Integration

Antiderivatives, or indefinite integrals, are the reverse process of differentiation. When given a derivative, finding an antiderivative means recovering the original function up to an arbitrary constant. This process is governed by the Fundamental Theorem of Calculus, which connects differentiation and integration in a profound way.

Fundamental Theorem of Calculus

This theorem bridges differentiation and integration by asserting that integration can be reversed by differentiation. In practice, it allows us to recover an original function by integrating its derivative, and then applying an initial condition to determine the constant of integration, ensuring a unique solution.

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