Comparison Test for Improper Integrals In somecases, it is impossible to find the exact value of an improperintegral, but it is important to determine whether the integralconverges or diverges. Suppose the functions f and g arecontinuous and 0 ≤ g ( x ) ≤ f ( x ) on the interval [ a , ∞ ) . It canbe shown that if ∫ a ∞ f ( x ) d x converges, then ∫ a ∞ g ( x ) d x alsoconverges, and if ∫ a ∞ f ( x ) d x diverges, then ∫ a ∞ g ( x ) d x alsodiverges. This is known as the Comparison Test for improperintegrals. (a) Use the Comparison Test to determine whether ∫ a ∞ g ( x ) d x converges or diverges. (Hint: Use the fact that e − x 2 ≤ e − x for x ≥ 1 .) (b) Use the Comparison Test to determine whether ∫ 1 ∞ 1 x 5 + 1 d x converges or diverges. (Hint: Use the fact that 1 x 5 + 1 ≤ 1 x 5 for x ≥ 1 .)
Comparison Test for Improper Integrals In somecases, it is impossible to find the exact value of an improperintegral, but it is important to determine whether the integralconverges or diverges. Suppose the functions f and g arecontinuous and 0 ≤ g ( x ) ≤ f ( x ) on the interval [ a , ∞ ) . It canbe shown that if ∫ a ∞ f ( x ) d x converges, then ∫ a ∞ g ( x ) d x alsoconverges, and if ∫ a ∞ f ( x ) d x diverges, then ∫ a ∞ g ( x ) d x alsodiverges. This is known as the Comparison Test for improperintegrals. (a) Use the Comparison Test to determine whether ∫ a ∞ g ( x ) d x converges or diverges. (Hint: Use the fact that e − x 2 ≤ e − x for x ≥ 1 .) (b) Use the Comparison Test to determine whether ∫ 1 ∞ 1 x 5 + 1 d x converges or diverges. (Hint: Use the fact that 1 x 5 + 1 ≤ 1 x 5 for x ≥ 1 .)
Solution Summary: The author analyzes whether the improper integral displaystyle 'int' converges or not according to the comparison test.
Comparison Test for Improper Integrals In somecases, it is impossible to find the exact value of an improperintegral, but it is important to determine whether the integralconverges or diverges. Suppose the functions f and g arecontinuous and
0
≤
g
(
x
)
≤
f
(
x
)
on the interval
[
a
,
∞
)
. It canbe shown that if
∫
a
∞
f
(
x
)
d
x
converges, then
∫
a
∞
g
(
x
)
d
x
alsoconverges, and if
∫
a
∞
f
(
x
)
d
x
diverges, then
∫
a
∞
g
(
x
)
d
x
alsodiverges. This is known as the Comparison Test for improperintegrals.
(a) Use the Comparison Test to determine whether
∫
a
∞
g
(
x
)
d
x
converges or diverges. (Hint: Use the fact that
e
−
x
2
≤
e
−
x
for
x
≥
1
.)
(b) Use the Comparison Test to determine whether
∫
1
∞
1
x
5
+
1
d
x
converges or diverges. (Hint: Use the fact
that
1
x
5
+
1
≤
1
x
5
for
x
≥
1
.)
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
(Improper Integrals)
9.1) Consider whether the following Improper Integrals are convergent or divergent, if convergent find the value of the integral. (Ans. Divergent)
(Improper Integrals)
9.9) Consider whether the following Improper Integrals are convergent or divergent, if convergent find the value of the integral. (Ans. Convergent -1/2)
Improper Integrals
1 – Check if the following integrals are convergent or divergent
Write with simple and easiest way to solve this question with clear written question
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