Summary Properties of Definite Integrals | Calculus II Notes

1.2 Basic Properties of Definite Integrals
Algebraic properties:
1) Sa(f(x) g(x)(dx Saf(x) Sag(x)
= = +




2) SPCf(x)dX cSaf(x)dX,=
where c is constant


follow directly from the definition of Saf(x)dx + the
corresponding properties for sums and limits



eg. imo, (f(x) =
g(x,*))Ax

-im (f(xx*) AX g(x,*)AX) =




-eimf(x) ax 1gIX*AX
-!f(x)ax) 2)g(x,*(x) =




-(aPf(x)dx =(ag(X)dX

we build up more complicated functions from simpler ones using these rules:



1) the simplestfunction:f(x) 1
=




geometry.
SaP 1.dX=SadX
by
=
1(b -
a) b
=
-
a




by definition in


Saqx=reimoktY**a ee



=(2) e(b a)
= -




=b -
a




2) f(x) x
=




Sydy b" =b2 ta
a 1b a A
=
=
=-
=
- -
-




, star
Y ·
y
=




ill
A
b2
=




or
or
Alan-tar
a b

A a
=

A
a
=


A
1b2
=




·b
note:if f(x) x
=
an antiderivative is F(x) = x2 =




and (axdX F(b) =
-
F(a)