Part II: The Answers
542
Answers
901–1,001
921.
1
7
7
7
2
ln
− −
+
x
x
C
The given integral is
dx
x
x
7
2
−
∫
Begin by using the substitutions x = 7 sinθ and dx
d
= 7 cosθ θ :
7
7
7 7
7 1
7
2
2
2
cos
sin
s in
cos
sin
s in
cos
sin
cos
θ θ
θ
θ
θ
θ
θ
θ
θ
θ
θ
d
d
−
=
−
(
)
=
∫
∫
d d
d
d
d
C
θ
θ
θ
θ
θ
θ
θ
θ θ
θ
θ
∫
∫
∫
∫
=
=
=
−
+
cos
sin
cos
sin
csc
ln csc
cot
7
1
7
1
1
7
1
7
From the substitution x = 7 sinθ , you get x
7
= sinθ , from which you can deduce that
7
x
= cscθ and also that cotθ =
−
7
2
x
x
. (To deduce that cotθ =
−
7
2
x
x
, you can label
a right triangle using x
7
= sinθ and find the missing side of the triangle with the
Pythagorean theorem.) Using these values, you get the following solution:
1
7
7
7
2
ln
− −
+
x
x
C
922.
1
2 5
5
5 25
2
1
sin
−
( ) + −
)
(
+
x
x
x
C
Rewrite the integral with a radical sign:
1 5
1
5
2
2
−
=
− ( )
∫
∫
x dx
x
dx
Now begin by using the substitution x 5 = sinθ , or x = 1
5
sinθ , so that
dx
d
= 1
5
cosθ θ :