7.2 - MATH
1081
 Wednesday,
April
6
 


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Unformatted text preview: MATH
1081
 Wednesday,
April
6
 
 Chapter
7
–
Section
2
 
 SUBSTITUTION
 Homework
#11
(due
4/11):
 








 
 Clicker
Check­in:

Choose
any
letter
to
check
in
now.
 
 
 Section
7.1
#16,
20,
54,
60
 Section
7.2
#18,
24,
32,
40,
42
 When
we
evaluate
the
integral
 ∫ 2 x x − 5 dx ,
we
do
have
the
 2 6 ( ) option
to
multiply
the
function
out
and
then
evaluate
term
by
 term,
but
this
would
require
some
fairly
lengthy
algebra.
 
 
 6 7 12 2 Instead,
if
we
consider
that
 ∫ 2 x x − 5 dx = x − 5 + C ,
 7 what
happened
to
the
 2 x 
part
of
the
original
integrand?
 ( ) ( ) The
method
of
substitution
can
be
used
to
simplify
an
 integrand.

In
particular,
when
we
have
an
integrand
that
is
the
 result
if
the
chain
rule
for
differentiation,
substitution
will
 allow
us
to
detect
and
remove
the
derivative
of
the
“inside”
 function
to
make
a
simpler
integral
to
evaluate.
 
 
 Recall
the
chain
rule
for
differentiation:
 
 
 
 
 
 D ⎡ f ( g ( x ) ) ⎤ = f ' ( g ( x ) ) ⋅ g ' ( x ) .
 ⎣ ⎦ So,
 
 
 
 
 ∫ f ' ( g ( x )) ⋅ g ' ( x ) dx = f ( g ( x )) + C .
 To
see
how
this
works,
let’s
look
again
at
 ∫ 2 x x − 5 dx .
 2 6 ( ) 
 6 2 If
we
let
 u = x − 5 ,
then
we
have
 ∫ 2 x ( u ) dx .

Already
this
looks
 simpler,
but
there
are
now
2
variables,
which
is
a
problem.
 
 du 2 However,
notice
that
if
 u = x − 5 ,
then
 = 2 x 
or
 du = 2 x dx .
 dx 
 Substituting
this
in
as
well,
we
now
have
 ∫ u 6 du 
which
is
very
 simple
to
evaluate.

 
 7 17 12 So,
we
have
 ∫ 2 x x − 5 dx = ∫ u du = u + C = x − 5 + C .
 7 7 2 6 6 ( ) ( ) If
we
think
of
the
chain
rule
as

“FIRST
take
the
derivative
of
 the
outside
function,
THEN
take
the
derivative
of
what’s
 inside”,
then
this
technique
of
integration
by
substitution
 “FIRST
undoes
the
derivative
of
what’s
inside,
THEN
undoes
 the
derivative
of
the
outside
function”.
 
 
 The
key
is
to
look
for
an
“inside”
function
and
its
derivative.
 
 Example:

Evaluate
the
indefinite
integral.
 
 6x2 
 
 
 1.

 ∫ 3/ 2 dx 
 3 2x + 7 
 
 ey 
 
 
 2.

 ∫ dy 
 y 
 
 
 
 
 
 3.

 ∫ ( 3x − 9 ) x 2 − 6 x dx 
 ( ) 
 Example:

Evaluate
the
integral.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1.

 ∫ p ( p + 1) dp 
 5 −4 x 2.

 ∫ 2 dx 
 x +3 −4 x 3.
 ∫ dx 
 x+3 Clicker
Check­out:

Press
any
letter
to
check
out
now.
 
 Tomorrow
in
recitation:

Workshop
 
 Next
time
–
Section
7.3
 
 ...
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