Applied Calculus: UNIT III: Integral Calculus
Comparison Test for Improper Integrals
Explanation, Formula, Equation, Example and Solved Problems - Integral Calculus: Comparison Test for Improper Integrals
COMPARISON
TEST FOR IMPROPER INTEGRALS
Suppose that f and g are continuous functions with f(x) ≥ g(x) ≥ 0 for x≥ a.
If a∫∞ f(x) dx is convergent, then a∫∞g(x)
dx is convergent.
If a∫∞ g(x) dx is convergent, then a∫∞f(x) dx is convergent.
Example
101. Does the integral 
is convergent?
Solution:
We know that 1/x2
> 1/(ex+x2) > 0.
Here 1∫∞
dx/x2 is convergent by the p‒test, since p = 2 > 1.
Therefore by comparison
test, is convergent
Example
102. Determine if the following integral is convergent or divergent.
Solution: We
know that
So 3∫∞e‒xdx
is convergent. Therefore, by the Comparison test
is also convergent.
Example
103. Does the integral 1∫∞ dx/xex converge?
Solution:
We have 1/ex
> 1/xex > 0.
1∫∞ dx/ex
convergent
Therefore by comparison
test, the integral 1∫∞ dx/xe‒x converges
Example
104. Does the integral 1∫∞ dx/√[1+x3] converges
Solution:
We have 1/√x3
> 1/(√[1+x3]) > 0.
Here 1∫∞
dx/√x3 is convergent by the p‒test, since p = 3/2 > 1.
Therefor by comparison
test, the integral 1∫∞ dx/√[1+x3] converges.
Example
105. Does the integral 
converge?
Solution:
is
divergent by the p‒test, since p = 2/5 ≤ 1.
Therefore by comparison
test, the integral diverges.
Example
106. Does the integral 2∫∞ (2+ sinx)/(x‒1) converge?
Solution: We
have (2+ sin x)/(x‒1) > 1/x > 0
Note that 2∫∞
dx/x is divergent by the p‒test, since p = 1 ≤ 1.
Therefore by comparison
test, the integral 2∫∞ (2+ sin x)/(x‒1) diverges.
Example
107. Determine if the following integral is convergent or divergent.
Solution:
We now know that we
need to find a function that is larger than
cos2(x) / x2
We can use the fact
that 0 ≤ cos2(x) ≤ 1 to make the numerator larger (i.e. we'll
replace the cosine with something we know to be larger, namely 1). So,
converges since p = 2
> 1converges and so by the Comparison Test we know that
must also converge.
EXERCISE
13. Check whether the
following improper integral converges or diverges, if converges evaluate it.
Applied Calculus: UNIT III: Integral Calculus : Tag: Applied Calculus : - Comparison Test for Improper Integrals
Applied Calculus: UNIT III: Integral Calculus
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