Name:
Score:
10B Practice Quiz
Please return during Monday’s lecture. You may use whatever resources you like.
1. (5 points) Find the area between the
x
-
axis and the function
f
(
x
) =
(1 + ln(
x
))
2
x
on the interval [1
,
3].
Solution:
Finding the area between the
x
-
axis and the function on [1
,
3] is the same as the
following definite integral
3
1
(1 + ln(
x
))
2
x
dx
For this, try the substitution,
w
= 1 + ln(
x
). This means that
dw
= (1
/x
)
dx
. Substituting everything
in, we get
w
2
x
x
dw =
w
2
dw
Now applying the power rule, we get
w
2
dw = (1
/
3)
w
3
+
C
Now replacing the
w
with
x
we get
(1 + ln(
x
))
2
x
dx
= (1
/
3)
w
3
+
C
= (1
/
3)(1 + ln(
x
))
3
+
C
Now applying the first fundamental theorem of calculus, we get
3
1
(1 + ln(
x
))
2
x
dx = (1
/
3)((1 + ln(3))
3
)
-
1)
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2. (5 points) Compute the following indefinite integral
(1 + tan(
x
))
3
cos
2
(
x
)
dx
Solution:
This is a substitution problem. First since (1
/
cos
2
(
x
) = sec
2
(
x
), we can write
(1 + tan(
x
))
3
cos
2
(
x
)
dx =
(1 + tan(
x
))
3
)sec
2
(
x
) dx
Now let
w
= 1 + tan(
x
), implying that
dw
= sec
2
(
x
)
dx
. Now substituting everything in, we get
w
3
sec
2
(
x
)(1
/
sec
2
(
x
)) dw
=
w
3
dw
= (1
/
4)
w
4
+
C
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- Summer '10
- reed
- Math, 6W
-
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