Integration Using Tables ~ p. 1
J. Ahrens 2000-2006
INTEGRATION USING TABLES
Objective: Evaluate integrals using tables of integrals
It is frequently necessary to use substitution along with integral tables.
It may be necessary to use a formula more than once or to use two or more formulas.
Evaluate 2
2
2
0
x12
dx
x4
⎛⎞
+
⎜⎟
⎜⎟
+
⎝⎠
∫
• The closest formula is # 17:
1
22
du 1 u
tan C
aa
au
−
=
+
+
∫
•2
22
x12 8
1
x4 x4
+=+
++
• Applying #17 with a = 2: =
2
2
2
0
x12
dx
x4
⎛⎞
+
⎜⎟
⎜⎟
+
⎝⎠
∫2
2
0
8
1dx
x4
⎛⎞
+
⎜⎟
+
⎝⎠
∫
= 2
11
0
1x
x8 tan 24tan12
22
π
−−
⎡⎤
+=+=+
⎢⎥
⎣⎦
Evaluate 2
2
xdx
54x−
∫
• The closest formula is #34: 22
22 1
22
uuau
du a u sin C
22a
au
−⎛⎞
=
−−+ +
⎜⎟
⎝⎠
−
∫
• Using substitution: let u = 2x
u
du 2dx and x 2
⇒= =
2
2
22
u
x1
2
dx du
2
54x 5u
⎛⎞
⎜⎟
⎝⎠
=
−−
∫∫
• Using # 34 with :
2
a5=
2
2
22
u
11u
2du du
28
5u 5u
⎛⎞
⎜⎟
⎝⎠ =
−−
∫∫
= 21
1u 5 u
5u sin C
82 2 5
−
⎡⎤
−−+ +
⎢⎥
⎣⎦
