Antiderivative through a point

Problem

Find the antiderivative of $f(x) = 3x^2-2$ that passes through the point $(1,4)$.

Solution

The family of antiderivatives is $F(x) = x^3 - 2x + C.$ We impose passage through $(1,4)$: $F(1) = 1-2+C = 4$, whence $C = 5$. The required antiderivative is $\boxed{F(x) = x^3 - 2x + 5.}$

Links

Topics: Integral
Concepts: Indefinite integral · Antiderivative
Skills: Integrate · Solve equations
Exercise types: Integral computation