Defines an improper integral and breaks down the steps for solving improper integrals with either an infinite domain of integration or with an unbounded integrated.
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
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller dazyskiies. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $6.79. You're not tied to anything after your purchase.
