Q.Evaluate: $\int (x^3+x^2-x+1)\,dx$

Integration is linear: the integral of a sum (or difference) of two functions equals the sum (or difference) of their integrals, $\int[f(x)\pm g(x)]\,dx=\int f(x)\,dx\pm\int g(x)\,dx$, and a constant factor can always be pulled outside the integral sign, $\int k\,f(x)\,dx=k\int f(x)\,dx$. Both facts follow directly from the corresponding rules for derivatives: if $G_1'(x)=f(x)$ and $G_2'(x)=g(x)$, then $\frac{d}{dx}[G_1(x)\pm G_2(x)]=f(x)\pm g(x)$, so $G_1(x)\pm G_2(x)$ is a primitive of $f(x)\pm g(x)$. These properties let a complicated-looking integrand be broken into a sum of simpler pieces — each matched separately against the elementary formulae — and are used in essentially every integral in this chapter, however sophisticated the surrounding technique.