3. ∫ sin7 θ cos 5 θdθ = ∫ sin7 θ(1-sin2 θ)2dsinθ = ∫ u7(1

3.
R π/2
0
1
.
120
7.
R π/2
11.
R π/2
0
0
π
.
16
sin7 θ cos5 θdθ =
cos2 θdθ =
R π/2
0
R π/2
0
1
2
sin7 θ(1−sin2 θ)2 d sin θ =
(cos(2θ) + 1) dθ =
R π/2
sin2 x cos2 xdx =
0
t−t cos(2t)
dt
2
1
4
1
4
sin2 (2x)dx =
t2
4
h
t sin(2t)
4
R π/2
0
23.
R
R
tan2 xdx = (sec2 x − 1)dx = tan x − x + C.
27.
R π/3
0
117
.
8
5
4
tan x sec xdx =
R π/3
0
5
−
R
u7 (1−u2 )2 du =
θ π/2
2 0
2
tan x(1+tan x)d tan x =
=
t2
4
R √3
0
h
u8
8
− 2 u10 +
−
sin(4x)
32
10
u12
12
i1
= π4 .
(1 − cos(4x)) dx =
i
t sin2 tdt =
−
1
8
sin(2t)
dt
4
R
=
0
sin(2θ) +
13.
R
R1
−
t sin(2t)
4
5
h
x
8
−
cos(2t)
8
2
u (1+u )du =
h
u6
6
iπ/2
=
0
+ C.
+
u8
8
i √3
=
0
R
sin(8x) cos(5x)dx = 12 sin(13x) + sin(3x)dx = − cos(13x)
+ − cos(3x)
+ C.
26
6
R
2
2x
56. (a) u(−du) = −u2 = − cos
+ C.
2
R
2
2
(b) udu = u2 = sin2 x + C 0 . C 0 = C − 12 .
R
(c) 21 sin(2x)dx = − cos(2x)
+ C 00 . C 00 = C − 14 .
4
R
R
R
2
(d) sin x cos xdx = sin x sin x− cos x sin xdx ⇒ sin x cos xdx = sin2 x +C 000 . C 000 =
C 0.
π/4
R π/4
R π/4
sin(2x) 2
2
57. −π/4 (cos x − sin x)dx = −π/4 cos(2x)dx = 2 = 1.
41.
R
−π/4
61.
Rπ
π/2
2
π(sin x) dx =
π
2
Rπ
π/2
(1 − cos(2x))dx =
π
2
h
x−
sin(2x)
2
iπ
π/2
=
π2
.
4
h
iπ
Rπ
68. If m = n, 21 −π (1 − cos(2mx))dx = 12 x − sin(2mx)
= π. If m 6= n,
2m
h
iπ −π
n)x − cos(m + n)x)dx = 21 sin(m−n)x
− sin(m+n)x
= 0.
m−n
m+n
1
2
Rπ
−π
(cos(m −
−π
70. By 68.,
1
π
Rπ
−π
N
P
an sin(nx) sin(mx)dx =
n=1
1
1
π
Rπ
−π
am sin(mx) sin(mx)dx = am .
0
=