SL 5.5—Integration as anti-differentiation

For integer $n\ne-1$, $\int ax^n\,dx=ax^{n+1}/(n+1)+C$. An indefinite integral is a family of anti-derivatives; a boundary condition determines the constant $C$.

A definite integral $\int_a^b f(x)\,dx$ connects anti-derivatives with accumulated signed area. At SL, use technology after first writing the correct integral; for a region above the x-axis, the integral equals its geometric area.

Example

If $dy/dx=3x^2+x$ and $y=10$ when $x=1$, then $y=x^3+x^2/2+C$ and $10=1+1/2+C$, so $C=8.5$. Thus $y=x^3+x^2/2+8.5$.

Do not omit $C$ in an indefinite integral or apply the power formula to $x^{-1}$. A differential-equation growth model belongs to AHL, not this SL anti-differentiation Objective.