Trigonometric SubstitutionIntegrals involving 2 ...
Trigonometric Substitution Integrals involving
a2 - x2
Integrals
involving
p x
2
+
a2
Integrals involving
x2 - a2
Trigonometric Substitution
To solve integrals containing the following expressions;
p a2 - x2
p x2 + a2
p x2 - a2,
it is sometimes useful to make the following substitutions:
Expression q
a2 - x2
pa2 + x2
q x2 - a2
Substitution
x = a sin ,
-
2
2
or
= sin-1
x a
x = a tan ,
-
2
2
or
= tan-1
x a
x = a sec ,
0
<
2
or
<
3 2
or
= sec-1
x a
Identity 1 - sin2 = cos2 1 + tan2 = sec2
sec2 - 1 = tan2
Note The calculations here are much easier if you use the substitution in
reverse:
x
=
a sin
as
opposed
to
= sin-1
x a
.
Annette Pilkington
Trigonometric Substitution
Trigonometric Substitution Integrals involving
a2 - x2
Integrals
involving
p x
2
+
a2
Integrals involving
Integrals involving a2 - x2
x2 - a2
We make
a2 - x2
the substitution
=
p a2
-
a2
sin2
x
= =
a sin , a| cos |
-
2
=a
2
cos (since
, dx = a cos d,
-
2
2
by choice. )
Example
Z x3
dx
4 - x2
Annette Pilkington
Trigonometric Substitution
Trigonometric Substitution Integrals involving
a2 - x2
Integrals
involving
p x
2
+
a2
Integrals involving
Integrals involving a2 - x2
x2 - a2
We make
a2 - x2
the substitution
=
p a2
-
a2
sin2
x
= =
a sin , a| cos |
-
2
=a
2
cos (since
, dx = a cos d,
-
2
2
by choice. )
Example
Z x3
dx
4 - x2
Let x = 2 sin , dx = 2 cos d,
4
-
x2
=
p 4
-
4
sin2
=
2
cos
.
Annette Pilkington
Trigonometric Substitution
Trigonometric Substitution Integrals involving
a2 - x2
Integrals
involving
p x
2
+
a2
Integrals involving
Integrals involving a2 - x2
x2 - a2
We make
a2 - x2
the substitution
=
p a2
-
a2
sin2
x
= =
a sin , a| cos |
-
2
=a
2
cos (since
, dx = a cos d,
-
2
2
by choice. )
Example
Z x3
dx
4 - x2
Let x = 2 sin , dx = 2 cos d,
4
-
x2
=
p 4
-
4
sin2
=
2
cos
.
R
x 3 dx 4-x 2
=R
8 sin3 (2 cos d) 2 cos
=R
8 sin3 d = R
8 sin2 sin d =
8 R (1 - cos2 ) sin d.
Annette Pilkington
Trigonometric Substitution
Trigonometric Substitution Integrals involving
a2 - x2
Integrals
involving
p x
2
+
a2
Integrals involving
Integrals involving a2 - x2
x2 - a2
We make
a2 - x2
the substitution
=
p a2
-
a2
sin2
x
= =
a sin , a| cos |
-
2
=a
2
cos (since
, dx = a cos d,
-
2
2
by choice. )
Example
Z x3
dx
4 - x2
Let x = 2 sin , dx = 2 cos d,
4
-
x2
=
p 4
-
4
sin2
=
2
cos
.
R
x 3 dx 4-x 2
=R
8 sin3 (2 cos d) 2 cos
=R
8 sin3 d = R
8 sin2 sin d =
8 R (1 - cos2 ) sin d.
Let w = cos , dw = - sin d,
Z 8
(1-cos2 ) sin
d
=
Z -8
(1-w 2)
dw
Z =8
(w 2-1) dw
=
8w 3 -8w +C
3
=
8(cos )3 3
- 8 cos + C
=
8(cos(sin-1 3
x 2
))3
- 8 cos(sin-1
x 2
)
+
C
.
Annette Pilkington
Trigonometric Substitution
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