The Mathematics of Financial Modelingand Investment Management

4-PrincipCalculus Page 133 Friday, March 12, 2004 12:39 PM

---

-------------

Principles of Calculus 133

As an immediate consequence of the fundamental theorem of calculus

we can now state that, given a primitive Fof a function f, the definite integral

b

∫ fx()dx

a

can be computed as

b

∫ fx()dx= Fb()– Fa()

a

All three properties—the linearity of the integration operation, the chain

rule, and the rule of integration by parts—hold for indefinite integrals:

hx()= af x()+ bg x()⇒ ∫ hx()dx= afx∫ ()dx+ bgx∫ ()dx

∫ f′()xgx()dx= f x()gx()– ∫fx()g′()xdx

y= g x()⇒∫fyd ()y= ∫ fx()g′()xdx

The differentiation formulas established in the previous section can now

be applied to integration. Exhibit 4.9 lists a number of commonly used

integrals.

EXHIBIT 4.9 Commonly Used Integrals

f(x)

∫ ()dx

fx Domain

xn (^1) n+ 1 n≠–1, R, x≠0 if n< 0
-------------x
n+ 1
xα ------------- (^1) x α+ 1 x> 0
α+ 1
sin x –cos x R
cos x sin x R
1 log x x> 0
x
ex ex R
f′() x log [f(x)] f(x) > 0
fx()