Antiderivatives and the Definite Integral

≈ 25 min

Integration reverses differentiation. An antiderivative of $f$ is a function $F$ with $F'=f$ . Because the derivative of a constant is zero, antiderivatives come in families:

\int 2x\,dx = x^2 + C

The power rule for integration raises the power by one and divides by the new power:

\int x^n\,dx = \frac{x^{n+1}}{n+1} + C \quad (n\neq -1)

A definite integral $\int_a^b f(x)\,dx$ measures the (signed) area under $f$ between $x=a$ and $x=b$ . The Fundamental Theorem of Calculus connects the two ideas: find any antiderivative $F$ , then subtract its values at the endpoints.

\int_a^b f(x)\,dx = F(b) - F(a)

Worked example. $\int_0^3 2x\,dx = \big[x^2\big]_0^3 = 3^2 - 0^2 = 9.$

Which of these is an antiderivative of $2x$ ?

$2$

$x^2 + C$

$2x^2 + C$

$\dfrac{x^2}{2}+C$

Evaluate $\int_0^3 2x\,dx$ .

Evaluate $\int_1^2 3x^2\,dx$ .

A reservoir fills at a rate of $r(t)=2t+3$ kilolitres per hour. How many kilolitres flow in during the first 4 hours, i.e. $\int_0^4 (2t+3)\,dt$ ?