Part II: The Answers
536
Answers
901–1,001
Set this expression equal to the given integral:
sec
s ec tan
sec
sec
3
3 x dx
x
x
xdx
x dx
∫
∫
∫
=
−
+
After evaluating sec x dx
∫
on the right side of the equation, you get
sec
s ec tan
sec
ln sec
tan
3
3 x dx
x
x
xdx
x
x
∫
∫
=
−
+
+
Add sec
3 x dx
∫
to both sides of the equation and then divide by 2 to get the solution:
sec
s ec tan
ln sec
tan
3
1
2
x dx
x
x
x
x
C
∫
=
+
+
)
(
+
912.
–sec x – tan x + C
The given integral is
dx
x
sin −
∫
1
Begin by multiplying the numerator and denominator of the integrand by the conjugate
of the denominator. Recall that the conjugate of (a – b) is (a + b). In this case, you multiply the numerator and denominator by (sin x + 1):
1
1
1
1
1
1
1
2
sin
( sin
)
(sin
)
(sin
)
sin
sin
x
dx
x
x
x
dx
x
x
−
=
−
+
+

 

 
=
+
−
∫
∫
1 1
dx
∫
Now use the identity sin
2 
x – 1 = –cos
2 
x and split apart the fraction to further simplify
the integrand:
sin
sin
sin
cos
sin
cos
cos
x
x
dx
x
x
dx
x
x
x
dx
+
−
=
+
−
= −
+




=
∫
∫
∫
1
1
1
1
2
2
2
2
− −
+
(
)
= −
+
(
)
= −
+
∫
∫
sin
cos cos
sec
tan sec
sec
(sec
tan
x
x
x
x dx
x
x
x dx
x
1
2
2
x x C
x
x C
)
sec
tan
+
= −
−
+