SL 5.10—Indefinite integration

Indefinite integration reconstructs a family of functions.

Reverse the derivative term by term and retain the arbitrary constant; the constant represents the unknown vertical shift.

Example

∫(2x+sin x)dx=x²−cos x+C.

Differentiate the answer and use any initial value to fix C.

Do not omit C when no initial condition has selected one antiderivative.

Formula set: $\int x^n\,dx=x^{n+1}/(n+1)+C$ for $n\ne-1$, $\int dx/x=\ln|x|+C$, $\int e^x\,dx=e^x+C$, $\int\sin x\,dx=-\cos x+C$, and $\int\cos x\,dx=\sin x+C$. For a linear composite, account for the inner gradient: $\int\cos(2x+3)\,dx=\tfrac12\sin(2x+3)+C$. Reverse-chain recognition gives $\int k g'(x)[f(g(x))]\,dx$ by substitution or inspection.