Identify the given function.

The function $f(x)$ is defined as an integral:

The Fundamental Theorem of Calculus, Part 1 states that if $F(x) = \int_{a}^{x} g(t) dt$, then $F'(x) = g(x)$.
In this case, $g(t) = t^3 e^t$.
Therefore, to find $\frac{d}{dx} f(x)$, we substitute $x$ for $t$ in the integrand.

Applying the theorem, we get:

The calculated value is $x^3 e^x$.
Option (a) is $e^x (x^3 + 3x^2)$.
Option (b) is $x^3 e^x$.
Option (c) is $a^3 e^a$.
Option (d) is इनमें से कोई नहीं (None of these).

The result matches option (b).

The final answer is $\text{(b)} x^{3} e^{x}$.