Answers and Explanations 523
Answers
801–900
881.
2
2
x
x
x C
sin
c os
+
+
The given integral is
cos xdx
∫
Begin by using the substitution w
x
=
so that w
2
= x and 2wdw = dx. This gives you the
following:
cos
c os
x dx
w
w dw
∫
∫
= 2
Now use integration by parts with u = 2w so that du = 2dw and let dv = cos w
 
dw so that
v = sin w:
2
2
2
2
2
w
wdw
w
w
w dw
w
w
w C
cos
s in
sin
sin
cos
∫
∫
=
−
=
+
+
Using the original substitution, w
x
=
, gives you the solution:
2
2
2
2
w
w
w C
x
x
x C
sin
cos
sin
c os
+
+
=
+
+
882.
x
x
x
x
x C
5
2
5
5
5
2
25
2
125
(ln )
ln
−
+
+
Here’s the given integral:
x
x dx
4
2
(ln )
∫
Begin by using integration by parts with u = (ln x)
2
so that du
x x
dx
= 2
1
(ln )
and
let dv = x
4 
dx so that v x
=
5
5
:
x
x dx x
x
x
x x
dx
x
x
x
x d
4
2
5
2
5
5
2
4
5
2
5
1
5
2
5
(ln )
(ln )
(ln )
(ln )
(ln )
∫
∫
∫
=
−
=
−
x x
To evaluate the new integral, use integration by parts again with u 1 = ln x so that
du
x
dx
1
1
=
and let dv = x
4 
dx so that v x
=
5
5
:
x
x
x
x
x
x
dx
x
x
x
x
x
5
2
5
5
5
2
5
4
5
2
5 5
5
1
5
2
25
2
25
(ln )
ln
(ln )
ln
−
−






=
−
+
∫
d dx
x
x
x
x
x
C
x
x
x
x
∫
=
−
+





 +
=
−
+
5
2
5
5
5
2
5
5
2
25
2
25 5
5
2
25
(ln )
ln
(ln )
ln
2 2
125
5
x C
+