Part II: The Answers
514
Answers
801–900
862.
x
x
x
x C
sec
ln sec
tan
−
+
+
The given integral is
x
x
x dx
sec tan
∫
Use integration by parts with u = x so that du = dx and let dv = sec x tan xdx so that
v = sec x:
x
x
x dx x
x
xdx
x
x
x
x C
sec tan
sec
sec
sec
ln sec
tan
∫
∫
=
−
=
−
+
+
863.
−
+
+
x
x
x C
cos( )
sin( )
5
5
1
25
5
The given integral is
x
x dx
sin( )
5
∫
Begin by using integration by parts with u = x so that du = dx and let dv = sin(5x)
 
dx
so that v
x
= −
cos( )
5
5
:
x
x dx
x
x
x dx
x
x
x
sin( )
c os( )
cos( )
cos( )
sin(
5
1
5
5
1
5
5
1
5
5
1
25
5
∫
∫
= −
+
= −
+
) ) + C
864.
−
+
+
x
x
x C
cot
ln sin
The given integral is
x
xdx
csc
2
∫
Begin by using integration by parts with u = x so that du = dx and let dv = csc
2 
x
 
dx so
that v = –cot x:
x
xdx
x
x
xdx
x
x
x C
csc
c ot
cot
cot
ln sin
2
∫
∫
= −
+
= −
+
+
865.
–x cos x + sin x + C
The given integral is
x
xdx
sin
∫