Integral

Definition: If $F(x)$ is an antiderivative of $f(x)$, then the \emph}integral} of the function $f(x)$ over the interval $[a,b]$ is defined by $$\int_a^b f(x)dx=F(b)-F(a).$$

Remark 1: In stead of $F(b)-F(a)$ we also write $[F(x)]_{x=a}^{x=b}$.

Remark 2: For calculating an integral of a function $f(x)$ we only need one (of the infinitely many ) antiderivatives $F(x)+c$. For simplicity we normally choose $c=0$.


Example 1
$$\int_1^2 (3x^2+2) dx=[x^3+2x]_{x=1}^{x=2}=(2^3+2\cdot 2)-(1^3+2\cdot 1)=12-3=9.$$

Example 2
$$\int_0^1 4e^x dx=[4e^x]_{x=0}^{x=1}=(4e^1)-(4e^0)=4e-4=4(e-1).$$

Example 3
$$\int_3^8 \frac{1}{\sqrt{x+1}} dx=[2\sqrt{x+1}]_{x=3}^{x=8}=(2\sqrt{9})-(2\sqrt{4})=6-4=2.$$