It looks like there's a typo or a small mistake in the expression you've provided. I'll assume you're asking about an integral involving a vector field \( \mathbf{v}(t) \), a gradient operator \( \nabla \), and an exponential function. However, the exact expression "gradint^edx" isn't clear.

Could you please clarify your question? Specifically, do you mean something like:

1. \( \int \mathbf{v}(t) \cdot \nabla e^{x} \, dx \)?
2. \( \int \nabla \cdot \mathbf{v}(t) \, dx \)?
3. Something else?

If you provide more details, I can offer a more precise solution!

Explore the integral involving a vector field 𝐯(t) and the gradient of an exponential function e^x. This problem delves into vector calculus concepts and techniques in integral calculus. Suitable for undergraduate students and those interested in vector calculus.

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