Answers and Explanations 507
Answers
801–900
844.
2
3
2
1
(sin
sin )
−
Rewrite the given integral:
x
x
dx
x
x
dx
0
1
3 2
1 2
0
1
3 2
1
1
∫
∫
+
(
) =
+
(
)
cos
c os
Begin by using the substitution u = 1 + x
3/2
so that du
x dx
= 3
2
1 2
, or 2
3
du
x dx
=
. Notice
you can find the new limits of integration by noting that if x = 1, then u = 1 + 1
3/2
= 2,
and if x = 0, then u = 1 + 0
3/2
= 1. With these new values, you find that
2
3
2
3
2
3
2
1
1
2
1
2
cos
s in
(sin
sin )
u du
u
∫
=
=
−
845.
7
8
8 7
(tan )
x
C
+
Rewrite the given integral:
tan sec
(tan ) sec
x
xdx
x
xdx
7
1
2
7
2
∫
∫
=
Then use the substitution u = tan x so that du = sec
2 
x
 
dx:
u du u
C
x
C
1 7
8 7
8 7
8 7
7
8
∫
=
+
=
+
(tan )
846.
1
π
π
cos x
C
+
Here’s the given integral:
sin π
x
x
dx
( )
∫ 2
Use the substitution u x
x
= =
−
π π
1 so that du
x dx
x
dx
= −
= −
−
π
π
2
2
, or −
=
1
1
2
π
du x
dx:
sin
sin
cos
cos
π
π
π
π
π
x
x
dx
u du
u C
x
C
( ) = −
=
+
=
+
∫
∫
2
1
1
1