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Calculus: Differentiation and Integration, 1st Edition
book

Calculus: Differentiation and Integration, 1st Edition

by ICFAI
April 2024
Intermediate to advanced content levelIntermediate to advanced
710 pages
20h 50m
English
Pearson India
Content preview from Calculus: Differentiation and Integration, 1st Edition
11-26  Calculus – Differentiation and Integration
Example 11.21 Evaluate
dx
x
1 2
0
+
cos
.
p
Solution:
I
dx
x
=
+
1 2
0
cos
p
(1)
=
+
=
+
=
+
dx
dx
x
x
x
1 2
11
1 2
2
1 2
0
0
cos( )
cos
cos
co
( )
p
p
p
by property
ss
.
x
dx
0
p
Thus,
I dx
x
x
=
+
2
1 2
0
cos
cos
p
(2)
Adding Eqs. (1) and (2), we obtain
2
1
1 2
2
1 2
1
1 2
2
1 2
0 0
I dx dx
x
x
x
x
x
x
=
+
+
+
=
+
+
+
cos
cos
cos
cos
cos
cos
p p

=
+
+
=
dx
dx dx
x
x
0
0 0
1 2
1 2
p
p p
(by property 5)
cos
cos
==
[ ]
=
x
0
p
p
(by FTC)
.
=
+
=I
dx
x
1 2
2
0
cos
.
p
p
Example 11.22 Evaluate
ln .
3
3
3
3
+
x
x
dx
Solution:  Let
g x
x
x
( ) ln=
+
3
3
Now consider
g x
x
x
x x
( ) ln
ln( ) ln( )
=
+
= +
3
3
3 3
Integration 11-27
= +
( )
=
+
=
ln( ) ln( )
ln ...
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Publisher Resources

ISBN: 9789353948221