1.2. THE FUNDAMENTAL THEOREM 29
3.
Z
1
1
x
6
C x
5
C x
4
C x
3
C x
2
C x C1
/
dx
4.
Z
3
p
3
3
p
3
tan
1
.x/ dx
5.
Z
2
0
.
sin.x/ C sin.2x/ Csin.3x/
/
dx
6.
Z
a
a
x
2n
dx
7.
Z
a
a
x
2nC1
dx
8.
Z
a
a
sin.x/ C tan
1
.x/C x
5
dx
Problem 1.38 Classify each of the following functions as being odd, even, or neither.
1. y D sin.x/
2. y D cos.x/
3. y D tan
1
.x/
4. y D ln.x
2
C 1/
5. y D x ln.x
2
C 1/
6. y D e
x
2
=2
7. y D x
2
C x C1
8. y D sin.x
2
/, and
9. p.x
2
/, where p.x/ is any polynomial.
Problem 1.39 Suppose that f .x/ is an even function. Prove that y D x f .x/ is an odd func-
tion.
Problem 1.40 Suppose that f .x/ is an odd function. Prove that y D x f .x/ is an even func-
tion.
Problem 1.41 Suppose that f .x/ is a function and g.x/ is an even function. Is f .g.x// an
even function? Demonstrate your answer is correct.
Problem 1.42 Find b when
Z
b
0
x
2
dx D 14
30 1. INTEGRATION, AREA, AND INITIAL VALUE PROBLEMS
Problem 1.43 Find b when
Z
b
0
x
4
dx D 1
Problem 1.44
If
Z
0
sin.x/ dx D 2;
what is the smallest possible value for ?
Problem 1.45 Find a constant c so that the area bounded by y D c and y D x
2
is exactly
4 units
2
.
Problem 1.46 Compute the slope m so that the area bounded by the curves y D x
3
and y D
mx is exactly 4 units.
Problem 1.47 Find the largest possible value of
Z
b
a
cos.x/ dx.
Problem 1.48 Compute
Z
5
5
x
3
ln
x
2
C 1
dx
Problem 1.49 Suppose
F .x/ D
Z
x
0
e
t
2
dt
What is
F
0
.x/
?
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