Answers and Explanations 539
Answers
901–1,001
From x = secθ , you can deduce that tanθ =
−
x
2
1. (To deduce that tanθ =
−
x
2
1,
you can label a right triangle using x
1
= secθ and then use the Pythagorean theorem
to find the missing side.) Therefore, the solution becomes
tanθ + =
− +
C
x
C
2
1
Note that you can also solve the problem with the substitution u = x
2
– 1.
917.
− +
+
1
2
x
x
C
The given integral is
1
1
2
2
x
x
dx
+
∫
Begin with the trigonometric substitution x = tanθ to get dx
d
= sec
2
θ θ , which gives
you the following:
sec
tan
t an
sec
tan
sec
sec
tan
cos
2
2
2
2
2
2
2
1
1
θ θ
θ
θ
θ
θ
θ
θ
θ
θ
θ
θ
d
d
d
+
=
=
=
∫
∫
∫
c cos
sin
cos
sin sin
cot csc
csc
2
2
1
θ
θ
θ
θ
θ
θ
θ
θ
θ θ
θ
d
d
d
C
∫
∫
∫
=
=
= −
+
From x = tanθ , you can deduce that cscθ =
+
1
2
x
x
. (To deduce that cscθ =
+
1
2
x
x
,
you can label a right triangle using x
1
= tanθ and then find the missing side using the
Pythagorean theorem.) Therefore, you get the following solution:
−
+ = − +
+
cscθ C
x
x
C
1
2
918.
1
16
16
2
x
x
C
−
+
The given integral is
1
16
2
2
x x
dx
−
∫
Begin by using the substitution x = 4 secθ to get dx
d
= 4 sec tan
θ
θ θ . Substituting
these values into the original integral, you have