Part II: The Answers
546
Answers
901–1,001
Now use a trigonometric identity and simplify:
sec
tan
sec
sec
sec
ln sec
tan
2
2
0
4
2
2
0
4
0
4
1
θ
θ
θ
θ
θ
θ
θ θ
θ
θ
π
π
π
+
=
=
=
+
∫
∫
∫
d
d
d
( (
)
=
+
−
+






=
+ −
+
=
+
0
4
4
4
0
0
2 1
1 0
2 1
π
π
π
ln sec
tan
ln sec
tan
ln
ln
ln
927.
x
x C
2
1
1
− −
+
−
sec
Here’s the given integral:
x
x
dx
2
1
−
∫
Begin with the substitution x = secθ so that dx
d
= sec tan
θ
θ θ :
sec
sec
sec tan
tan tan
tan
2
2
2
1
θ
θ
θ
θ θ
θ
θ θ
θ θ
−
=
=
∫
∫
∫
d
d
d
Next, use an identity and integrate to get
sec
t an
2
1
θ
θ
θ θ
−
(
) =
− +
∫
d
C
From the substitution x = secθ , you can deduce that sec
−
=
1 x θ and that tanθ =
−
x
2
1
to get the solution:
x
x C
2
1
1
− −
+
−
sec
928.
3
2
The given integral is
1
1
2
2
1
2
x x
dx
−
∫
Use the trigonometric substitution x = secθ so that dx
d
= sec tan
θ
θ θ . You can find the
new limits of integration by noting that if x = 1, then 1 = secθ and θ = 0 and that if x = 2,
then 2 = secθ and θ π
= 3
. With these new values, you get