Contradictory results for Berezin integral Say $$u = (u_1, \dots, u_{2n}) = (\xi_1, \eta_1, \dots \xi_n, \eta_n)\tag{1}$$ is a vector of Grassmann variables. For an antisymmetric  bosonic matrix $A$ we know that
$$
\int e^{\frac{1}{2} \sum_{a,b} A_{ab}(\xi_a\eta_b - \eta_a\xi_b)} \prod_{i=1}^n d\xi_i d\eta_i = \det(A) \ .\tag{2}
$$
Now, I need to calculate $\mathbb E\left[e^{\langle t,u\rangle}\right]$, with $t = (t_1,\dots,t_{2n})$ is bosonic; that is,
$$
\mathbb E\left[e^{\langle t,u\rangle}\right] = \int e^{\langle t,u\rangle} \ e^{\frac{1}{2} \sum_{a,b} A_{ab}(\xi_a\eta_b - \eta_a\xi_b)} \prod_{i=1}^n d\xi_i d\eta_i \ .\tag{3}
$$
On the one hand, we know $$e^{\langle t,u\rangle} = 1 + \langle t,u\rangle,\tag{4}$$ in which case
$$
\mathbb E\left[e^{\langle t,u\rangle}\right] = \int e^{\frac{1}{2} \sum_{a,b} A_{ab}(\xi_a\eta_b - \eta_a\xi_b)} \prod_{i=1}^n d\xi_i d\eta_i + \sum_{j=1}^{2n} t_j \int u_j \ e^{\frac{1}{2} \sum_{a,b} A_{ab}(\xi_a\eta_b - \eta_a\xi_b)} \prod_{i=1}^n d\xi_i d\eta_i $$
$$= \det(A) + \sum_{j=1}^{2n} t_j \langle u_j\rangle = \det(A)\tag{5}
$$
since $\langle u_j\rangle = 0$. On the other hand, $\xi_a \eta_b$ behaves as a commuting variable (since it's a product of two Grassmann). So my guess is that I can put the variables $u_i$ above in the exponent, obtaining
$$
\mathbb E\left[e^{\langle t,u\rangle}\right] = \int e^{\sum_{j=1}^{2n} t_j u_j + \frac{1}{2} \sum_{a,b} A_{ab}(\xi_a\eta_b - \eta_a\xi_b)} \prod_{i=1}^n d\xi_i d\eta_i \ .\tag{6}
$$
Using the last property here, we know this is equal to $\det(A) \ e^{-t^T A^{-1} t}$.
So, which one is the correct result for $\mathbb E\left[e^{\langle t,u\rangle}\right]$? $\det(A)$ or $\det(A) \ e^{-t^T A^{-1} t}$?
 A: It is only true that $\exp(a) = 1+a$ if $a$ is a Grassman variable. A product of two Grassman variables is not Grassman.
For example, consider (with Grassman $a,b,c,d$)
$$\exp(ab+cd) = 1 + ab + cd + \frac{1}{2}(2abcd) + ...$$
In fact, this must be the case or else we would have
$$\int \exp\left(-\frac{1}{2}A_{ab}(\xi_a\eta_b - \eta_b\xi_a) \right)\prod d\xi d\eta
= \int 1 - \frac{1}{2}A_{ab}(\xi_a\eta_b - \eta_b\xi_a) \prod d\xi d\eta$$
which is zero.
I believe the source of your issues is that the change of variables $\eta \to \eta + t$ only makes sense if $t$ is in the Grassman algebra - if it is an ordinary complex number, this operation is not defined.
(see the Appendix A.3 of this paper).
However, the Grassmans are still a $\mathbb{C}$-vector space, so $tu$ is well defined for $t\in\mathbb{C}$, $u$ Grassman.
In that case, since the argument of the exponential is linear in all Grassman variables, $\exp(\langle t,u \rangle) = 1 + \langle t,u \rangle$ is correct, and $\mathbb{E}(e^{tu}) = \det A$.
