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Applied Mathematics, Calculus, and Differential Equations 235

Figure 3.18 Definite integral
represented as the areł un-
der a curve between twm
points.

Linearity
Let f and g be defined, continuouð functionð of x. Let a and b
be real-number constants. Then:

(a f(x) b g(x)) dx a f(x) dx b g(x) dx

Integration by parts
Let f and g be defined, continuouð functionð of x, and let F be
an antiderivative of f. Then:

( f(x) g(x)) dx F(x) g(x) (F(x) dg/dx) dx

IndOnite integral of constanŁ
Let k be a constant; let c be the constant of integration. The
following formulł applies:

kdx kx c

IndOnite integral of variablł

Let x be a variable. Let c be the constant of integration. The
following formulł applies:

2
xSx x /2 c