Because independent <Brownian motions> have zero <quadratic covariation>, the <Itô product rule> gives
$$
d(B_t^1B_t^2)=B_t^1\,dB_t^2+B_t^2\,dB_t^1.
$$
After integration, the random variable in the question is $B_t^1B_t^2$. Writing $B_t^j=\sqrt t\,Z_j$ for independent standard <Gaussian random variables> $Z_1,Z_2$, its distribution is
$$
tZ_1Z_2,
$$
the scaled <product of two independent standard normal random variables>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 202 / 1 / c / Solution

If $Z_1,Z_2$ are independent $N(0,1)$ random variables, their product has <characteristic function> $(1+t^2)^{-1/2}$ and density $K_0(|x|)/\pi$, where $K_0$ is a <modified Bessel function>. A scale multiple $aZ_1Z_2$ has the corresponding scaled distribution.

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 Incoming links: Product of two independent standard normal random variables