Answers and Explanations 533
Answers
901–1,001
Now use the substitution u = sec x so that du = sec x tan x
 
dx. This gives you
u
du u u C
x
x C
2
3
3
1
3
1
3
−
(
) = − +
=
−
+
∫
sec
sec
906.
−
+
+
1
3
3
(csc ) csc
x
x C
Begin by factoring out cot x and using an identity:
cot csc
cot csc cot
csc
c sc cot
2
3
2
1
x
xdx
x
x
x dx
x
x
x dx
∫
∫
∫
=
=
−
(
)
Now use the substitution u = csc x so that du = –csc x cot x
 
dx, or –du = csc x cot x
 
dx.
This gives you the following:
−
−
( ) = − + +
= −
+
+
∫ u
du
u
u C
x
x C
2
3
3
1
3
1
3
(csc ) csc
907.
2
3
2
7
3 2
7 2
(sin )
(sin )
x
x
C
−
+
Here’s the given integral:
cos
sin
3 x
xdx
∫
Begin by factoring out cos x and using a trigonometric identity to write cos
2
x in terms
of sin x:
cos
sin
cos
sin cos
sin
sin cos
3
2
2
1
x
xdx
x
x
x dx
x
x
x dx
∫
∫
∫
=
=
−
(
)
Now use u-substitution: u = sin x so that du = cos x. Using these values in the integral
gives you
1
1
2
3
2
7
2
3
2
2
1 2
1 2
5 2
3 2
7 2
−
( )
=
−
( )
=
−
(
)
=
−
+
=
∫
∫
∫
u
udu
u u du
u
u
du
u
u
C
x
(sin ) )
(sin )
3 2
7 2
2
7
−
+
x
C