Answers and Explanations 475
Answers
701–800
762.
1
2 1
2
+
(
)
x
Rewrite the radical in the given function using an exponent:
y
x
x
x
x
=
− +
(
)
=
− +
(
)
(
)
−
−
tan
tan
1
2
1
2
1 2
1
1
Use the derivative formula for tan
−1 
x along with the chain rule:
′ =
+ − +
(
)
−
+
(
) ( )
(
)
= +
−
+
+ +

 


−
y
x
x
x
x
x
x
x
x
1
1
1
1 1
2
1
2
1
1
2 1
1
2
2
2
1 2
2
2
2
/
  − +

 

 
1 1
2
x
x
Now the fun algebra simplification begins:
′ = +
−
+
+ +

 

  − +

 

 
=
+
−
+

 


y
x
x
x
x
x
x
x
x
x
1
1
2 1
1
1 1
1
2 2
2 1
2
2
2
2
2
2  
+
−
+

 

 
=
+
−
+
+
−
+
(
)
=
+
−
+
+
1
1
1
2 1
1
1
1
2 1
1
2
2
2
2
2
2
2
2
2
x
x
x
x
x
x
x
x
x
x
x
x
x
( (
) − +
(
))
(
=
+
−
+
(
) + −
(
)
=
+
(
)
x
x
x
x
x
x
x
x
1
1
2 1
1
1
2 1
2
2
2
2
2
763.
π
2
Here’s the initial problem:
2
1
2
0
1
x
dx
+
∫
Because you have the antiderivative formula
1
1
2
2
1
x a
dx a
x
a
C
+
=
( ) +
−
∫
tan
, you can
simply apply that here to find the solution: