Q.Verify that $\displaystyle\int(3x^2+2x)\,dx = x^3+x^2+C$ by differentiating the answer.

If $\frac{d}{dx}F(x)=f(x)$, then $F$ is an antiderivative of $f$, and the indefinite integral

$\int f(x)\,dx=F(x)+C$ collects the ENTIRE family of antiderivatives (differing by a constant

$C$), since the derivative of any constant is zero. Linearity ($\int[af+bg]\,dx=a\int f\,dx+b\int g\,dx$) allows term-by-term integration of a sum. Every computed integral should be checked by

differentiating the answer and confirming it reproduces the original integrand -- the definitive

verification method used throughout this chapter.