Answers and Explanations 529
Answers
801–900
Now use the substitution u = tan x so that du = sec
2 
x
 
dx. You can also change the limits
of integration by noting that if x = π
3
, then u =
=
tan π
3
3, and if x = 0, then u = tan 0 = 0.
With these new values, the integral becomes
u
u du
u u du
u
u
3
2
0
3
3
5
0
3
4
6
0
3
4
6
4
1
4
6
3
4
3
6
0
4
+
(
) =
+
(
)
=
+






=
( ) + ( ) −
∫
∫
+ +









 



 
= +
=
0
6
9
4
27
6
27
4
6
897.
−
−
+
1
6
1
4
6
4
(cot )
(cot )
x
x
C
The given integral is
cot
sin
3
4
x
x
dx
∫
Begin by rewriting the integral and factoring out csc
2 
x:
cot
sin
cot csc
cot csc csc
3
4
3
4
3
2
2
x
x
dx
x
xdx
x
x
x dx
∫
∫
∫
=
=
Now use an identity to get the following:
cot
cot
c sc
3
2
2
1
x
x
x dx
+
(
)
∫
Next, use the substitution u = cot x to get du = –csc
2
x
 
dx so that
−
+
(
) = − −
(
)
= − −
+
= −
−
+
∫
∫
u u
du
u u du
u
u
C
x
x
C
3
2
5
3
6
4
6
4
1
6
4
1
6
1
4
(cot )
(cot )
898.
2
3
Here’s the given integral:
cos
3
0
2
x dx
π
∫
Begin by factoring out cos x and using a trigonometric identity to write cos
2 
x in
terms of sin x:
cos cos
sin
cos
2
0
2
2
0
2 1
x
xdx
x
xdx
π
π
∫
∫
=
−
(
)