Part II: The Answers
518
Answers
801–900
872.
x
x
x
C
cos
−
− −
+
1
2
1
Here’s the given integral:
cos
−
∫
1 x dx
Use integration by parts with u = cos
−1 
x so that du
x
dx
= − −
1
1
2
and let dv = 1dx
so that v = x:
cos
c os
−
−
∫
∫
=
+
−
1
1
2
1
x dx x
x
x
x
dx
Next, use the substitution w = 1 – x
2
so that dw = –2x
 
dx, or −
=
1
2
dw x dx:
x
x
x
x
dx x
x
w
dw
x
x
w dw
x
x
cos
c os
cos
cos
−
−
−
−
−
+
−
=
−
=
−
=
∫
∫
∫
1
2
1
1
1 2
1
1
1
2
1
1
2
− − ( ) +
=
− −
+
−
1
2
2
1
1 2
1
2
w
C
x
x
x
C
cos
873.
x
x
x
C
sin ( )
−
+
−
+
2
1 5
1
5
1 25
The given integral is
sin ( )
−
∫
1 5x dx
Use integration by parts with u = sin
−1
(5x) so that du
x
dx
=
−
1
1 5
5
2
( )
and let dv = 1dx
so that v = x:
sin ( )
s in ( )
−
−
∫
∫
=
−
−
1
1
2
5
5
5
1 25
x dx x
x
x
x
dx
Next, use the substitution w = 1 – 25x
2
so that dw = –50x
 
dx, or −
=
1
10
5
dw
x dx:
x
x
x
x
dx
x
x
w
dw
x
x
w
sin ( )
sin ( )
sin ( )
−
−
−
−
−
=
+
=
+
∫
∫
1
2
1
1
5
5
1 25
5
1
10
1
5
1
10
− −
−
−
∫
=
+ ( ) +
=
+
−
+
1 2
1
1
2
2
1
5
1
10
2
5
1
5
1 25
dw
x
x
w
C
x
x
x
C
sin ( )
sin ( )