
11-36 Calculus – Differentiation and Integration
Example 11.33 Determine if
x x
x
dx
2
2
1
+
∫
∞
converges or diverges.
Solution: For x > 0, we have x
2
+ x > x
2
⇒
+
>
⇒
+
>
⇒
+
>
⇒
+
∫
>
∫
∞ ∞
x x
x
x x
x
x x
x
x
x x
x
x
2
2
2
2
2
2
2
1 1
1
1
1
1
.
However, we have
1
1
x
∞
∫
diverges by p-test.
Thus, by comparison test
x x
x
dx
2
2
1
+
∫
∞
diverges.
Limit Comparison Test
Assume that f and g are continuous and f > 0 and g > 0 on [a, ¥). If
lim
( )
( )
,
x
f x
g x
c
→∞
= where c is some
positive finite number, then both f x dx
a
( )
∞
∫
and
g x dx
a
( )
∞
∫
converge or both diverge.
Proof: Suppose that
g x dx
a
( )
∞
∫
converges.
Also assume that
lim
( )
( )
.
x
f x
g x
c
→∞
=
So there exists b > a such that for all
x b
f x
g x
c≥ −