Find the function f given that the slope of the tangent line...

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.
$$f^{\prime}(x)=\frac{2}{x}+1 ;(1,2)$$

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Key Concepts

Indefinite Integration

Indefinite integration is the process of finding a function whose derivative is a given function. It is often used to 'reverse' differentiation by computing antiderivatives, resulting in a general solution that includes an arbitrary constant. This concept is fundamental when reconstructing a function from its rate of change.

Initial Value Problems

An initial value problem involves solving a differential equation along with an initial condition that specifies the value of the unknown function at a particular point. This additional information allows for the determination of the constant of integration, leading to a unique solution.

Antiderivatives

Antiderivatives are functions whose derivatives equal the given function. Finding an antiderivative is equivalent to integrating the function, and it is a central concept in solving differential equations where the derivative of the function is known but the function itself is not.

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