nanni (arn437) – Assignment 15 – guntel – (54940)
1
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4
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001
10.0points
Evaluate the integral
I
=
integraldisplay
π/
4
0
sec
2
x
(1

4 sin
x
)
dx .
1.
I
= 5 + 4
√
2
2.
I
=

3

4
√
2
3.
I
=

3

4
√
2
4.
I
= 5 +
4
√
2
5.
I
=

3 +
4
√
2
6.
I
= 5

4
√
2
correct
Explanation:
Since
sec
2
x
{
1

4 sin
x
}
= sec
2
x

4 sec
x
parenleftBig
sin
x
cos
x
parenrightBig
,
we see that
I
=
integraldisplay
π/
4
0
{
sec
2
x

4 sec
x
tan
x
}
dx .
But
d
dx
tan
x
= sec
2
x ,
while
d
dx
sec
x
= sec
x
tan
x .
Consequently,
I
=
bracketleftBig
tan
x

4 sec
x
bracketrightBig
π/
4
0
= 5

4
√
2
.
002
10.0points
Determine the integral
I
=
integraldisplay
x
2
+ 1
x

1
dx .
1.
I
=
x
2
+
x

ln

x

1

+
C
2.
I
=
1
2
x
2
+
x
+ 2 ln

x

1

+
C
correct
3.
I
=
x
2
+
x

ln

x
+ 1

+
C
4.
I
=
1
2
x
2
+
x

2 ln

x

1

+
C
5.
I
=
1
2
x
2

x
+ 2 ln

x
+ 1

+
C
6.
I
=
x
2

x

ln

x
+ 1

+
C
Explanation:
After division
x
2
+ 1
x

1
=
(
x
2

1) + 2
x

1
=
x
2

1
x

1
+
2
x

1
=
x
+ 1 +
2
x

1
.
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 Fall '10
 Gualdini
 Calculus, Classless InterDomain Routing, Chinese surname, ln x

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