Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states:
\[\int^b_af(x) \; \mathrm{d} x = F(b) - F(a)\] where  \(F(x)\) is the antiderivative of \(f(x)\).

There are two types of integrals:

• Indefinite integral: No limits involved, produces an algebraic formula with an unknown constant.  The indefinite integral has the notation, and terminology:

\[ \underbrace{\int \overbrace{f(x)}^{\mbox{integrand}} \; \mathrm{d} x}_{\mbox{integral}}
= F(x) + \underbrace{C}_{\mbox{constant of integration}}\]

• Definite integral: Always evaluated between limits to produce a measurable quantity.  This is the same as the net area under the curve between the limits.

For example find the indefinite integrals:

1. \(\displaystyle \int\frac{2x^3 + 3x + 1}{x} \; \mathrm{d} x\) 
2. \( \displaystyle \int (\cos x - e^x) \; \mathrm{d} x\)
3. \( \displaystyle \int \sqrt{x^3} + x^{\frac{2}{3}} \; \mathrm{d} x\)