Answers and Explanations 535
Answers
901–1,001
910.
−
+
−
+
1
5
2
7
1
9
5
7
9
cos
c os
cos
x
x
x C
The given integral is
sin cos
5
4
x
xdx
∫
Begin by factoring out sin x and using a trigonometric identity to write sin
2
x in terms
of cos x:
sin cos sin
sin
cos sin
cos
cos sin
4
4
2
2
4
2
2
4
1
x
x
x dx
x
x
x dx
x
x
x
∫
∫
= (
)
=
−
(
)
d dx
∫
Now use the substitution u = cos x, which gives you du = –sin x
 
dx, or –du = sin x
 
dx.
Using these values in the integral gives you
−
−
(
)
= −
−
+
(
)
= −
−
+
(
)
= −
−
+
∫
∫
∫
1
1 2
2
5
2
7
2
2 4
2
4
4
4
6
8
5
7
u
u du
u u u du
u
u u du
u
u
u
9 9
5
7
9
9
1
5
2
7
1
9





 +
= −
+
−
+
C
x
x
x C
cos
c os
cos
911.
1
2
sec tan
ln sec
tan
x
x
x
x
C
+
+
)
(
+
To evaluate this integral, you can use integration by parts. You may want to first
rewrite sec
3 
x as sec
2 
x sec x (rewriting isn’t necessary, but it may help you see things
a bit better!):
sec
s ec sec
3
2
x dx
x
x dx
∫
∫
=
Now use integration by parts with u = sec x so that du = sec x tan x
 
dx and let
dv = sec
2 
x
 
dx so that v = tan x. This gives you the following:
sec sec
sec tan
tan (sec tan )
sec tan
tan sec
2
2
x
xdx
x
x
x
x
x dx
x
x
x
x
∫
∫
=
−
=
−
d dx
∫
Now use an identity to get
sec tan
tan sec
sec tan
sec
s ec
sec tan
s
x
x
x
xdx
x
x
x
xdx
x
x
−
=
−
−
(
)
=
−
∫
∫
2
2
1
e ec
sec
3 x dx
xdx
∫
∫
+