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Math 104 -‐ Yu
Math 104 – Calculus 8.4 Trigonometric Subs=tu=ons
Math 104 -‐ Yu
Trigonometric subs=tu=on • Some=mes it is very helpful to use trigonometric iden==es to simplify involving radical expressions.
• It can be used to evaluate integrals containing the following expressions:
• Don’t forget to change the limits of the definite integrals!
1.
pa
2 � x
2
2.
pa
2+ x
2or
1
a
2+ x
2
3.
px
2 � a
2
Math 104 -‐ Yu
Square root of a2 -‐ x2
Nicolas Fraiman Math 104
Integrals with square root of a2-x2 Let
Then
pa
2 � x
2=
pa
2 � a
2sin
2✓
=
qa
2(1� sin
2✓)
=
pa
2cos
2✓
= a| cos ✓|
x = a sin ✓
Nicolas Fraiman Math 104
Integrals with square root of a2-x2 Let
Then
We haveSince Then Thus
pa
2 � x
2=
pa
2 � a
2sin
2✓
=
qa
2(1� sin
2✓)
=
pa
2cos
2✓
= a| cos ✓|
x = a sin ✓
✓ = arcsinx
a
�a x a
pa
2 � x
2= a cos ✓
�⇡2 ✓ ⇡
2 and cos ✓ > 0
We have ✓ = arcsin
x
a
Since �a x a,
Then �⇡
2
✓ ⇡
2
and cos ✓ � 0.
Thus
pa
2 � x
2= a cos ✓.
Math 104 -‐ Yu
Example
Nicolas Fraiman Math 104
Examples1. Evaluate
Z p2
1
dx
x
2p4� x
2
Math 104 -‐ Yu
Example 2. Find the area of the unit disk.
Math 104 -‐ Yu
Tips
Nicolas Fraiman Math 104
Remark
To evaluate a definite integral we either undo the change of variable, or find the limits in the new variable θ.
To undo the change of variable we use the reference triangle to find formulas for functions of θ.
To find the new limits we use inverse trig. functions.
Math 104 -‐ Yu
Square root of a2 + x2
Nicolas Fraiman Math 104
Integrals with square root of a2+x2 Let
Then
pa
2 + x
2 =p
a
2 + a
2 tan2 ✓
=q
a
2(1 + tan2 ✓)
=pa
2 sec2 ✓
= a| sec ✓|
x = a tan ✓
Nicolas Fraiman Math 104
Integrals with square root of a2+x2 Let
Then
We haveThen Thus
pa
2 + x
2 =p
a
2 + a
2 tan2 ✓
=q
a
2(1 + tan2 ✓)
=pa
2 sec2 ✓
= a| sec ✓|
x = a tan ✓
✓ = arctanx
a
�⇡2 ✓ ⇡
2 and sec ✓ > 0
pa
2 + x
2 = a sec ✓
We have ✓ = arctanx
a
Since x can be any real number,
Then �⇡
2< ✓ <
⇡
2and sec ✓ � 1.
Thuspa
2 + x
2 = a sec ✓.
Math 104 -‐ Yu
Example
Nicolas Fraiman Math 104
Examples3. Find
!
!
!
4. Evaluate Z 1
0
dx
(x2 + 4)3/2
Z1
1 + x
2dx
Math 104 -‐ Yu
Square root of x2 -‐ a2
Nicolas Fraiman Math 104
Integrals with square root of x2-a2 Let
Then
We haveIf then soIf then so
px
2 � a
2 =pa
2 sec2 ✓ � a
2
=pa
2(sec2 ✓ � 1)
=pa
2 tan2 ✓
= a| tan ✓|
x = a sec ✓
x � a
✓ = arcsecx
a
x �a
⇡2 ✓ ⇡
0 ✓ ⇡2
px
2 � a
2 = a tan ✓
px
2 � a
2 = �a tan ✓Nicolas Fraiman Math 104
Integrals with square root of x2-a2 Let
Then
We haveIf then soIf then so
px
2 � a
2 =pa
2 sec2 ✓ � a
2
=pa
2(sec2 ✓ � 1)
=pa
2 tan2 ✓
= a| tan ✓|
x = a sec ✓
x � a
✓ = arcsecx
a
x �a
⇡2 ✓ ⇡
0 ✓ ⇡2
px
2 � a
2 = a tan ✓
px
2 � a
2 = �a tan ✓
We have ✓ = sec
�1 x
a
, note that |x| � a
1. If x � a then 0 ✓ <
⇡2 ,p
x
2 � a
2= a tan ✓.
2. If x �a then
⇡2 < ✓ ⇡,p
x
2 � a
2= �a tan ✓.
Math 104 -‐ Yu
Example 5. Evaluate
Zdxp
9x2 � 4.
Math 104 -‐ Yu
Summary
2/19/2014
2
Math 104 – Rimmer8.3 Trig. Substitution
2 23. x a!
L ct se ex a "= sec arcsecx
x aa
" "
= ⇒ =
0 or 2 2
# #" " #$ < < $
or x a x a< ! >
Assume 0.a >
2 2 tan for x a a x a"! = >
Quadrant 1 Quadrant 2.
0 but t for 0 and for 2 2
so we c
tan 0 ta
an't drop the absolute value gn
n 0
si
a# #
" " #" "> $ <% $ < $
( )22 2 2secx a a a"! = !
2 2 2seca a"= !
( )2 2sec 1a "= !
2 2tana "=
tana "=
sec secx ax
a" "= ⇒ =
"
x
a
2 2x a!
2 2 tan for x a a x a"= !! < !
Math 104 – Rimmer8.3 Trig. Substitution
2 21. a x! L nt se ix a "=
( )22 2 2 sina x a a "! = !
2 2 2sina a "= !
( )2 21 sina "= !
2 2cosa "=
2 2 cosa x a "! =
sin sinx ax
a" "= ⇒ =
"
xa
2 2a x!
2 23. x a! L ct se ex a "=
( )22 2 2secx a a a"! = !
2 2 2seca a"= !
( )2 2sec 1a "= !
2 2tana "=
2 2 tanx a a "! =
sec secx ax
a" "= ⇒ =
"
x
a
2 2x a!
2 22. a x+ L nt te ax a "=
( )22 2 2 tana x a a "+ = +
2 2 2tana a "= +
( )2 21 tana "= +
2 2seca "=
2 2 seca x a "+ =
tan tanx ax
a" "= ⇒ =
"
x
a
2 2a x+
2 2
# #"
!$ $ 2 2
# #"
!$ $
02
#"$ <
2
#" #< $
2 2 tanx a a "!! =
for x a>
for x a< !
�⇡
2< ✓ <
⇡
2
x � a
x �a
Math 104 -‐ Yu
Comple=ng squares
Nicolas Fraiman Math 104
Completing squares6. Evaluate
!
!
Hint: Write
Z 2
1
dxp4x� x
2
4x� x
2 = (�4 + 4x� x
2) + 4
= 4� (x� 2)2
Math 104 -‐ Yu
Remember u-‐subsitu=on
Nicolas Fraiman Math 104
Remember u-substitution!
!
• Evaluate
!
!
• Don’t use a trig. substitution if a u-substitution is easier.
Zxp
x
2 + 4dx