21.3. Evaluate the integral dx/[(x² + 1)(x² + 4)] by closing the contour (a) in the UHP and (b) in the LHP.
21.3. Evaluate the integral dx/[(x² + 1)(x² + 4)] by closing the contour (a) in the UHP and (b) in the LHP.
Chapter3: Functions
Section3.3: Rates Of Change And Behavior Of Graphs
Problem 1SE: Can the average rate of change of a function be constant?
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The question is about complex analysis and Calculus of Residues.
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![TESTTUE of f(Z)=
21.3. Evaluate the integral f dx/[(x2 + 1)(x² + 4)] by closing the contour
(a) in the UHP and (b) in the LHP.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F94dc01b6-0881-4fad-b8f8-3d9bb8d43f15%2Fae6c0a5e-b80a-4d09-8cce-05957df508ed%2Fy3zelwv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:TESTTUE of f(Z)=
21.3. Evaluate the integral f dx/[(x2 + 1)(x² + 4)] by closing the contour
(a) in the UHP and (b) in the LHP.
![Example 21.2.2. Let us now consider a more complicated integral:
x2 dx
(x2 + 1)(x² + 4)²
which turns into f, z2 dz/[(z² +1)(z² +4)²]. The poles in the UHP are at z = i and
= 2i. The former is a simple pole, and the latter is a pole of order 2. Thus,
22
1
Res[f(i)] = lim
|(z
i)
(z – i)(z + i)(z²+ 4)².
18i
22
1
Res[f(2i)] =
(2 – 1)! " dz (2 – 2i)².
22
d
lim
1)! z→2i dz
(z2 + 1)(z + 2i)² (z – 2i)².
d
lim
z→2i dz
[(22 + 1)(z + 2i)².
72i'
and
x² dx
5
2ni
(x2 + 1)(x² + 4)2
18i
72i
36
Closing the contour in the LHP would yield the same result as the reader is urged
to verify.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F94dc01b6-0881-4fad-b8f8-3d9bb8d43f15%2Fae6c0a5e-b80a-4d09-8cce-05957df508ed%2Fcd6st7p_processed.png&w=3840&q=75)
Transcribed Image Text:Example 21.2.2. Let us now consider a more complicated integral:
x2 dx
(x2 + 1)(x² + 4)²
which turns into f, z2 dz/[(z² +1)(z² +4)²]. The poles in the UHP are at z = i and
= 2i. The former is a simple pole, and the latter is a pole of order 2. Thus,
22
1
Res[f(i)] = lim
|(z
i)
(z – i)(z + i)(z²+ 4)².
18i
22
1
Res[f(2i)] =
(2 – 1)! " dz (2 – 2i)².
22
d
lim
1)! z→2i dz
(z2 + 1)(z + 2i)² (z – 2i)².
d
lim
z→2i dz
[(22 + 1)(z + 2i)².
72i'
and
x² dx
5
2ni
(x2 + 1)(x² + 4)2
18i
72i
36
Closing the contour in the LHP would yield the same result as the reader is urged
to verify.
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