Millersville University Name Answer Key
Department of Mathematics
MATH 211, Homework 1
January 23, 2006
1 Page 535, Exercise 48
Find all solutions to
sin x
4=−
1
2.
If x
4=−
π
6or x
4=−
5π
6then sin x
4=−
1
2. However additional solutions exists when angles of more than
one revolution of the origin are allowed. Thus we can summarize the solutions as
x
4=−
π
6+ 2nπ
−
5π
6+ 2nπ
x=−
2π
3+ 8nπ
−
10π
3+ 8nπ
where nis an integer. We can also express the solutions as a list as follows,
...,−
58π
3,−
50π
3,−
34π
3,−
26π
3,−
10π
3,−
2π
3,14π
3,22π
3,38π
3,46π
3. . . ,
2 Page 542, Exercise 32
Evaluate the integral, Z2x
9 + 9x4dx.
Z2x
9 + 9x4dx =Z2x
9(1 + x4)dx
=1
9Z2x
1 + x4dx
=1
9Z2x
1 + (x2)2dx
Integrate by substitution with
u=x2
du = 2x dx
thus
1
9Z2x
1 + (x2)2dx =1
9Z1
1 + u2du
=1
9tan−1u+C
=1
9tan−1(x2) + C.