It is possible to integrate this function? I  heard that  that there are some unintegratable functions and I want to as if this one is not one of them?
\begin{equation}
\large \int \frac{t}{t+1}dt
\end{equation}
Got this by trying to solve another function and I need to check if this is not a dead end. If it is I will just have to find another way.  It is likely that I just lack some skill and knowledge  to solve it despite it`s simple appearance.

Use $u = t+1$. Then $t = u-1$ and $du = dt$, so that 
$$
\int \frac t{t+1} dt = \int \frac{u-1}u du = \int 1 du - \int \frac 1u du = u - \log u + C = t+1 - \log(t+1) + C.
$$
"Unintegrable functions" are not functions with no primitive, they're just functions that we don't know how to compute the primitive by hand, i.e. they're not "expressible in terms of elementary functions".
Hope that helps,
