Answers and Explanations 569
Answers
901–1,001
Next, complete the square on the expression x
2
– x + 1:
x
x
x
x
x
2
2
2
1
1
4
1 1
4
1
2
3
4
− + =
− +
(
) + −
= −
( ) +
Splitting up the integral gives you
1
3
1
1
3
2
3
1
1
3
1
1
3
2
3
1
2
3
4
2
2
x
x
x x
dx
x
dx
x
x
+
+
−
+
− +










=
+
+
−
+
−
( ) +
∫
∫
∫
∫
dx
Using the substitution u x
= − 1
2
, you get du = dx and u
x
+ =
1
2
so that
−
+
( ) = −
1
3
1
2
1
3
u
x and −
+ = −
+
1
3
1
2
1
3
2
3
u
x
. Using these values gives you
1
3
1
1
3
2
3
1
2
3
4
1
3
1
1
3
1
2
3
2
2
2
x
dx
x
x
dx
x
dx
u
u
+
+
−
+
−
( ) +
=
+
+
−
+
+






∫
∫
( )
2 2
2
2
2
1
3
1
1
3
3
4
1
2
1
3
2
∫
∫
∫
∫
∫
=
+
+
−
+
+
+






dx
x
dx
u
u
du
u
du
( )
To evaluate the second integral, 1
3
3
4
2
−
+
∫
u
u
du, use the substitution w u
= +
2
3
4
so that
dw = 2u
 
du, or −
=−
1
2
dw
u du. This substitution gives you
−
=−
+
= −
+ +
= −
− + +
∫
1
6
1
1
6
1
6
3
4
1
6
1
2
2
w
dw
w C
u
C
x x
C
ln
ln
ln
Therefore, the solution is
1
3
1
1
3
3
4
1
2
1
3
2
1
3
1 1
6
2
2
2
2
x
dx
u
u
du
u
du
x
x
+
+
−
+
+
+






=
+ −
−
∫
∫
∫
( )
ln
ln
x x
u
C
x
x
x
+ +











 +
=
+ −
− + +
−
−
1 1
2
2
3
2
3
1
3
1 1
6
1
3
3
1
1
2
tan
ln
ln
tan
2 2 1
3
x
C
−





 +