Answers and Explanations 513
Answers
801–900
x
x dx x
x
x dx
x
x
x
cos( )
sin( )
sin( )
sin( )
sin( )
4
1
4
4
1
4
4
1
4
4
1
4
4
∫
∫
= (
) −
=
− ∫ ∫
=
+
+
dx
x
x
x C
1
4
4
1
16
4
sin( )
cos( )
859.
xe e C
x
x
− +
Here’s the given integral:
xe dx
x
∫
Use integration by parts with u = x so that du = dx and let dv = e
x 
dx so that v = e
x
. Using
the integration by parts formula gives you the following:
xe dx xe
e dx
xe e C
x
x
x
x
x
∫
∫
=
−
=
− +
860.
1
2
2
1
4
2
x
x
x C
cosh( )
sinh( )
−
+
Here’s the given integral:
x
x dx
sinh( )
2
∫
Use integration by parts with u = x so that du = dx and let dv = sinh(2x)
 
dx so that
v
x
= 1
2
2
cosh( ). Using the integration by parts formula gives you the following:
x
x dx x
x
x dx
x
x
sinh( )
cosh( )
cosh( )
cosh( )
sin
2
1
2
2
1
2
2
1
2
2
1
4
∫
∫
= (
) −
=
−
h h( )
2x C
+
861.
( )
ln
(ln )
3 6
6
1 6
6
3
3
2
+ −
The given integral is
x dx
x
6
0
3
∫
Use integration by parts with u = x so that du = dx and let dv = 6
x 
dx so that v
x
= 6
6
ln
.
Using the integration by parts formula gives you the following:
x dx x
dx
x
x
x
x
x
x
6
6
6
1
6
6
6
6
6
6
3 6
0
3
0
3
0
3
2
0
3
∫
∫
=
−
=
−






=
ln
ln
ln
(ln )
( )
3 3
3
2
2
3
3
2
6
6
6
0
1
6
3 6
6
1 6
6
ln
(ln )
(ln )
( )
ln
(ln )
−
− −








 


 
=
+ −