$\int_1^x (1+\frac{1}{t})^t \, dt = ex - \frac{e}{2}\log(x) + O(1)$

Question

I want to show this, but I'm not really sure where to get started. Integrating by parts hasn't really gotten me anywhere useful, and I'm not really sure what else to try.

Answer 1

$$(1+t^{-1})^t=\exp\left(t\ln\left(1+\frac1t\right)\right)=\exp\left(1-\frac{1}{2t}+O(t^{-2})\right) =e\left(1-\frac1{2t}+O(t^{-2})\right).$$ Integrate this from $1$ to $x$.