Answers and Explanations 537
Answers
901–1,001
913.
1
28
4
1
20
4
5
7
sec
s ec
x
x C
( )−
( )+
Here’s the given integral:
tan ( )sec ( )
3
5
4
4
x
x dx
∫
Begin by factoring out sec(4x)
 
tan(4x):
tan ( )sec ( ) sec( )tan( )
2
4
4
4
4
4
x
x
x
x dx
(
)
∫
Then use a trigonometric identity to arrive at
sec ( )
sec ( ) sec( )tan( )
2
4
4
1
4
4
4
x
x
x
x dx
−
(
)
(
)
∫
Now use the substitution u = sec(4x) so that du = 4 sec(4x)
 
tan(4x)
 
dx, or
1
4
4
4
du
x
x dx
= sec( )tan( ) . Using these substitutions, you now have
1
4
1
1
4
1
4 7
5
1
28
4
1
20
2
4
6
4
7
5
7
u
u du
u u du
u
u
x
−
( )
=
−
(
)
=
−

 

 
=
−
∫
∫
sec ( )
s sec ( )
5 4x C
+
914.
ln x
x C
2
4
2
+ + +
The given integral is
dx
x
2
4
+
∫
Start by using the substitution x = 2 tanθ so that dx
d
= 2
2
sec θ θ . Using these
substitutions in the integral, you get the following:
2
2
4
2
4
1
2
4
2
2
2
2
2
2
2
sec
( tan )
sec
tan
sec
sec
s
θ
θ
θ
θ
θ
θ
θ
θ
θ
+
=
+
(
)
=
=
∫
∫
∫
d
d
d
e ec
sec
sec
ln sec
tan
2
2
θ
θ
θ
θ θ
θ
θ
d
d
C
∫
∫
=
=
+
+