Example:
� ∞
10
dx
√
x3 + 3
�∞
We could have used a trig substitution to compute 0 √xdx
in the previous
2 +10
example. We can use the limit comparison method to determine whether an
integral is finite even if we’re unable
to find an antiderivative.
�∞
For instance, we can’t evaluate 10 √xdx
. But because:
3 +3
1
1
1
∼
√
= √ = 3/2
x
x3 + 3
x3
we know that:
�
∞
10
dx
∼
√
=
x3 + 3
�
∞
10
dx
x3/2
and so we know that the integral converges to some finite value.
1
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18.01SC Single Variable Calculus��
Fall 2010 ��
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