Integral of inverse Laplace transform
Hint.
As $\hat f(s) = \frac{1}{1+e^s}$ the residues for $\hat f(s)$ are located at $1+e^s = 0$ or over the imaginary axis at $(2k+1)\pi i,\ \ k\in \mathbb{Z}$ and $Res[\hat f((2k+1)\pi i)] = -1$ then
$$ \hat f(s) =-\lim_{n\to \infty}\sum_{k=0}^{k=n}\frac{2s}{s^2+(2k+1)^2\pi^2} $$
also
$$ f(t) = -2\lim_{n\to\infty}\sum_{k=0}^{k=n}\cos\left((2k+1)\pi t\right) $$
Follows the graphics for $\int_0^T f(t)dt$ with $n = 20$
