The vehicle is said to navigate along the curved pathF(x,y) = ∫ex^2[2x2 * y - x * y2 + y]dx.Which of the following paths is equivalent to the above given path?

To find an equivalent path to the given function $F(x,y) = \int e^{x^2} [2x^2 y - xy^2 + y] dx$, we need to evaluate the integral and understand its implications. Here are the steps to approach this problem:

1. Break Down the Problem

We have an integral representing a path, and our task is to find another expression or function that represents the same path. We need to first compute the integral.

2. Relevant Concepts

This is an integral calculus problem where we need to integrate a function with respect to $x$ while treating $y$ as a constant. The function provided is:

$F(x,y) = \int e^{x^2} (2x^2 y - xy^2 + y) \, dx$

3. Analysis and Detail

To compute $F(x,y)$:

1. Identify the parts of the integrand:

   • $e^{x^2}$ is a function of $x$.
   • The polynomial $(2x^2y - xy^2 + y)$ contains both $x$ and $y$.

2. Compute the integral term by term. The integral can be broken into three parts:

$F(x,y) = \int e^{x^2} \cdot 2x^2 y \, dx - \int e^{x^2} \cdot x y^2 \, dx + \int e^{x^2} \cdot y \, dx$

Each of these integrals could be solved either using integration by parts or looking up standard integrals of the form $\int e^{x^2} \, dx$.

4. Verify and Summarize

This computation could lead to quite complex results, and if typical paths are provided, we can numerically check their equivalence to the obtained $F(x,y)$.

Path functions usually follow certain properties like continuity and differentiability. To determine if another path representation is equivalent, we can check the first derivatives or evaluate them over a range of $x$ to see if they yield the same $F(x,y)$.

Final Answer

Unfortunately, without the specific options of paths provided, we cannot state which is equivalent. The final expression of $F(x,y)$ would need to be compared against potential candidates to identify equivalence.