CK12 Calculus - Single Variable

We can also use this substitutionmethodto evaluatedefiniteintegrals.If we attachlimitsof integrationto
our first example,we couldhavea problemsuchas

The methodstill works.However, we havea choiceto makeoncewe are readyto use the Fundamental
Theoremto evaluatethe integral.

Recallthat we foundthat for the indefiniteintegral.At this point,we
couldevaluatethe integralby changingthe answerbackto or we couldevaluatethe integralin But
we needto be careful.Sincethe originallimitsof integrationwerein , we needto changethe limitsof
integrationfor the equivalentintegralin Hence,

where

IntegratingProductsof Functions

We are not able to statea rule for integratingproductsof functions, but we can get a rela-
tionshipthat is almostas effective.Recallhow we differentiateda productof functions:

So by integratingboth sideswe get

or

In orderto rememberthe formula,we usuallywriteit as

We refer to this methodas integrationby parts.The followingexampleillustratesits use.
Example2: