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 fat any point $(x, f(x))$ is $f^{\prime}(x)$ and that the graph of $f$ passes through the given point.
$$f^{\prime}(x)=e^{x}+x ;(0,3)$$

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Key Concepts

Constant of Integration

When integrating a function to solve a differential equation, an arbitrary constant, known as the constant of integration, appears. This constant is determined using the initial condition provided in the problem, ensuring that the solution satisfies the given conditions.

Initial Value Problems

An initial value problem includes an additional piece of information, typically the value of the unknown function at a specific point, which is used to determine the constant of integration. This condition ensures a unique solution to the differential equation.

Differential Equations

Differential equations involve relationships between a function and its derivatives. In this context, the problem is about finding a function whose derivative is given, which is a common type of ordinary differential equation where the function can be determined by integrating its derivative.

Integration

Integration is the mathematical process of finding the antiderivative, which reverses differentiation. Since the derivative of the function is provided, integrating that derivative yields the original function up to an arbitrary constant.

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