2. In general you have $\int_{a}^{x} f(t)dt = F(x)-F(a)$ by the fundamental theorem of calculus. This is a function which you can graph.
Now you must use the partial fraction theorem to get that $\frac{2t}{t^3+1} = -\frac{2}{3(t+1)}+\frac{2t+2}{3(t^2-t+1)}$