Part II: The Answers
538
Answers
901–1,001
From the substitution x = 2 tanθ , you get x
2
= tanθ , from which you can deduce that
x
2
4
2
+ = secθ . (To deduce that x
2
4
2
+ = secθ , you can label a right triangle using
x
2
= tanθ and find the missing side using the Pythagorean theorem.) Using these
values, you get
ln sec
tan
ln
ln
θ
θ
+
+ =
+ + +
=
+ + +
C
x
x C
x
x C
2
2
4
2
2
4
2
915.
π
8
1
4
−
Here’s the given integral:
x
x
dx
2
2
0
2 2
1 −
∫
Begin by using the substitution x = sinθ so that dx
d
= cosθ θ . You can find the new
limits of integration by noting that if x = 2
2
, then
2
2
= sinθ so that π θ
4
= , and if x = 0,
then 0 = sinθ so that θ = 0:
sin cos
sin
sin
cos( )
2
2
0
4
2
0
4
0
4
1
1
2
1
2
1
2
θ
θ θ
θ
θ θ
θ θ
θ
π
π
π
d
d
d
−
=
=
−
(
)
=
∫
∫
∫
− −

 

 
=
−
( ) − −










= −
sin( )
sin
(
)
2
2
1
2 4
2
2
0 0
8
1
4
0
4
θ
π
π
π
π
916.
x
C
2
1
− +
Here’s the given integral:
x
x
dx
2
1
−
∫
Use the trigonometric substitution x = secθ so that dx
d
= sec tan
θ
θ θ :
sec
sec
sec tan
sec tan
tan
sec
tan
θ
θ
θ
θ θ
θ
θ θ
θ
θ θ
θ
2
2
2
2
1
−
=
=
=
+
∫
∫
∫
d
d
d
C