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.
The graph of f consists of line segments, as shown in the figure. Evaluate each definite Integral by using geometric formulas.
4
3.
(3, 2)
(4, 2)
2
(11, 1)
2
3
4
8
10 11
-2
-3
(8, -2)
(a)
-7f(x)
dx
(b)
(c)
8f(x) dx
11
(6) "
(d)
-4(x) dx
'11
(e) "2r(x) dx
10
4(x) dx
I just need this checked and corrected please
(11) Comparison Test for Improper Integrals. Suppose that functions f(x) and g(x) are continu-
ous and 0 ≤ g(x) ≤ f(x) on the interval [a, ∞]. It can be shown that if
f(x) da converges,
then
f(x) da also diverges.
Use the Comparison Test to determine whether the improper integral converges or diverges.
TO g(x) dx also converges, and if if 9 g(x) dx diverges, then
(b)
dx
r2+sinr
x² e-√x
dx
[₁²° √ ₁ + = =+d²x
(Ⓒ) 1+
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