Part II: The Answers
544
Answers
901–1,001
924.
1
4
2
2
4
1
2
2
sec
−
−
−





 +
x
x
x
C
The given integral is
x
x
dx
2
3
4
−
∫
Begin with the trigonometric substitution x = 2 secθ so that dx
d
= 2 sec tan
θ
θ θ :
4
4 2
2
2
2
8
2
3
2
3
sec
s ec tan
( sec )
tan
sec tan
sec
θ
θ
θ θ
θ
θ
θ
θ θ
θ
−
(
)
=
(
)
=
∫
∫
d
d
1 1
2
1
2
1
2
2
2
2
2
2
2
tan
sec
sin
cos
cos
sin
θ
θ
θ
θ
θ
θ θ
θ θ
d
d
d
∫
∫
∫
=
(
)
=
Now use an identity to get
1
2
1
2
1
2
1
4
2
2
1
4
2
2
1
4
−
(
)
=
−
(
) +
=
−
(
) +
=
−
∫
cos( )
sin
sin cos
(
θ
θ
θ
θ
θ
θ
θ
θ
d
C
C
s sin cos )
θ
θ + C
From the substitution x = 2 secθ , you can deduce that sec
−
=
1
2
x θ , sinθ =
−
x
x
2
4 , and
cosθ = 2
x
. (To find the values of sinθ and cosθ , you can label a right triangle using
x
2
= secθ and find the missing side of the triangle with the Pythagorean theorem.)
Therefore, the solution becomes
1
4
2
4
2
1
4
2
2
4
1
2
1
2
2
sec
sec
−
−
−
−





 ( )


 


 
+
=
−
−





x
x
x
x
C
x
x
x
  + C
925.
1
4
1
1
2
2
4
sin
−
( ) +
−
(
) +
x
x
x
C
The given integral is
x
x dx
x
x
dx
1
1
2
4
2
−
=
−
∫
∫
( )