Answers and Explanations 527
Answers
801–900
890.
1
5
5
(sin )
x
C
+
The given integral is
sin cos
4 x
xdx
∫
Use the substitution u = sin x so that du = cos x
 
dx:
u du u C
x
C
4
5
5
5
1
5
∫
=
+ =
+
(sin )
891.
1
3
3
(tan )
x
C
+
The given integral is
tan sec
2
2
x
xdx
∫
Begin by using the substitution u = tan x so that du = sec
2 
x
 
dx:
u du u C
x
C
2
3
3
3
1
3
=
+ =
+
∫
(tan )
892.
−
−
+
1
2
1
18
9
cos
c os( )
x
x C
Here’s the given integral:
sin( )cos( )
5
4
x
x dx
∫
Begin by applying the identity sin( )cos( )
sin(
) sin(
)
A
B
A B
A B
=
− +
+ )
(
1
2
to the
integrand to produce the integral 1
2
5
4
5
4
sin(
) sin(
)
x
x
x
x dx
−
+
+
)
(
∫
. Now simplify
and integrate:
1
2
5
4
5
4
1
2
9
1
2
1
sin(
) sin(
)
sin
sin( )
cos
x
x
x
x dx
x
x dx
x
−
+
+
)
(
=
+
(
)
=
−
−
∫
∫
9 9
9
1
2
1
18
9
cos( )
cos
c os( )
x
C
x
x C
(
) +
= −
−
+
893.
sec x + C
Here’s the given integral:
sec tan
x
xdx
∫
There’s a simple solution for this function:
sec tan
sec
x
xdx
x C
∫
=
+