1.9 Trigonometric Substitutions
for integrals involving yan-xx.
substitute x a =
sine
or =arcsin(Xa)
for integrals involving Nex"
substitute x atant
=
or = arctan (Xa)
these substitutions make what's under the square root a perfectsquare;
↓ac -
x
=
a2
=
-
a sin
a(l-sint).
=
COS28+Sin 28= 1
a ICOS28
=
1acuStl
=
if=arcsin (a), then -2181 T
SO COSE >O
taking aco,a"-x2=acost
a+x G2+artan?
= =
=a) I tant)
+
S
I tan2 Sec2O
=
a2 Sec8
=
=
1aSecl
if = arctan (Xa), then-e 8 <Iz
SOCOSt>0+ sect >0
taking a 0,92+ x2 asect
=
ey/ x,- ↳
Putx sint
=
d x COS dO
=
S
-sircos'OdO:sirtde
1 -
Sin
I so
COS28: 1+coS28
=Since d8=J1-cos28'd Sin28=1-cOS28, 2
2
=St dO-ISCOS28dO
e
u 28
=
du 2d =
It 4 ScOs udu
= -
for integrals involving yan-xx.
substitute x a =
sine
or =arcsin(Xa)
for integrals involving Nex"
substitute x atant
=
or = arctan (Xa)
these substitutions make what's under the square root a perfectsquare;
↓ac -
x
=
a2
=
-
a sin
a(l-sint).
=
COS28+Sin 28= 1
a ICOS28
=
1acuStl
=
if=arcsin (a), then -2181 T
SO COSE >O
taking aco,a"-x2=acost
a+x G2+artan?
= =
=a) I tant)
+
S
I tan2 Sec2O
=
a2 Sec8
=
=
1aSecl
if = arctan (Xa), then-e 8 <Iz
SOCOSt>0+ sect >0
taking a 0,92+ x2 asect
=
ey/ x,- ↳
Putx sint
=
d x COS dO
=
S
-sircos'OdO:sirtde
1 -
Sin
I so
COS28: 1+coS28
=Since d8=J1-cos28'd Sin28=1-cOS28, 2
2
=St dO-ISCOS28dO
e
u 28
=
du 2d =
It 4 ScOs udu
= -
