Docsity
Docsity

Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity


Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan


Guidelines and tips
Guidelines and tips

Notes on Evaluate Integrals using Tables of Integrals - Calculus II | MATH 1920, Study notes of Calculus

Material Type: Notes; Class: Calculus II; Subject: Mathematics; University: Pellissippi State Technical Community College; Term: Unknown 2000;

Typology: Study notes

Pre 2010

Uploaded on 08/16/2009

koofers-user-axv
koofers-user-axv 🇺🇸

10 documents

1 / 2

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
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
⎡⎤
−−+ +
⎢⎥
⎣⎦
pf2

Partial preview of the text

Download Notes on Evaluate Integrals using Tables of Integrals - Calculus II | MATH 1920 and more Study notes Calculus in PDF only on Docsity!

Integration Using Tables ~ p. 1 J. Ahrens 2000-

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

0 2

x 12 dx

x 4

  • The closest formula is # 17:

1

2 2

du 1 u tan C a u a^ a

− = +

2

2 2

x 12 8 1 x 4 x 4

  • Applying #17 with a = 2: =

2 2

0 2

x 12 dx

x 4

2

0 2

1 dx

x 4

2 1 1

0

1 x x 8 tan 2 4 tan 1 2 2 2

π

Evaluate

2

2

x dx

5 −4x

  • The closest formula is #34:

2 2 2 2 1

2 2

u u a u du a u sin C 2 2 a a u

− ⎛^ ⎞

− ⎝^ ⎠

  • Using substitution: let u = 2x

u du 2dx and x 2

2

2

2 2

u

x (^1 ) dx du 2 5 4x 5 u

  • Using # 34 with :

2 a = 5

2

2

2 2

u

1 2 1 u du du 2 8 5 u 5 u

1 u 2 5 1 u 5 u sin C (^8 2 2 )

⎢ −^ −^ +^ ⎥+

Integration Using Tables ~ p. 2 J. Ahrens 2000-

Evaluate

3 x sin x dx

  • Using #84 with n = 3:

3 3 2 x sin x dx = − x cos x + 3 x cos x dx

  • Using #85 with n = 2:

2 2 x cos x dx = x sin x − 2 x sin x dx

  • Using #82: (^) ∫x sin xdx = sin x −x cosx
  • Combining all results:

3 x sin x dx

3 2 = − x cos x + 3x sin x − 6 sin x + 6xcosx +C