SL 5.5—Integration as anti-differentiation

Integration reverses differentiation up to a constant.

An antiderivative F satisfies F′=f; all antiderivatives differ by a constant because differentiation removes constants.

Example

∫(6x²−4)dx=2x³−4x+C; differentiating checks the result.

Integrate each term, include +C for an indefinite integral and use a condition to determine C.

An indefinite integral is a family, not one function until a condition is supplied.

A boundary condition selects the constant: if $dy/dx=3x^2+x$ and $y(1)=10$, then $y=x^3+x^2/2+C$ gives $C=8.5$. A definite integral accumulates signed change; when $f(x)>0$, technology can evaluate $\int_a^b f(x)\,dx$ as the area between the curve and the $x$-axis. Write the correct integral before calculating.