Summary Improper Integrals | Calculus II Notes

1.12 Improper Integrals
Definition:
an integral having either,an infinite limitofintegration or an unbounded integrand
eg. S8, ** and So

Improper integrals with infinite domain of integration:

a) ifthe integral Jaf(x)dx exists for all Ra
Sif(x) dx
GS,f(x)dx
=




when the limitexists and is finite


b) ifthe integral SPf(x)dxexists for all rcb

Sf(x)dX eim
= SPf(x)dx
n -0
-




When the limitexists and is finite


c) ifthe integral S."f(x)dx exists for all rai
f(x)
Sf(x)dX
e Sif(x)dx+ im S, dx
=




when both limits existand are finite Canyc can be uses)


example Sa
firstintegrate to a finite domain


Sa, =arctances ("a
=arctan(R) -
arctan(a)


take the limitas R =
0




S,* R)*, **
=




-Rim/arctan (R) -arctan()

=I-arctan(a)

Improper integrals with undounded integrand:

a) if (f(x) dx exists for all a <t<b,

Sf(x)dx eim
=

Sf(x)dX
t -> at


when the limitexists and is finite


b) if Saf(x) dxexists for all a CTcb