6. Use residue theory to compute each of the following real integrals: cos() dr
solve a and c only
1. Use the residue theorem to evaluate the following integrals: cos θ (2+sin θ) 2n 0 0 1+sin2 θ 2x-0820, de (C) Jo 5-4cos
Use the theory of residues to evaluate the following real, definite integrals: 210 do Hint: integrate on the unit circle, z = elo; dz = izdo. Hint: find the residue in the upper half plane. ſº I dx 13+5 sin e dx b. So y2 + 4x +5 J-ox? +dx
use residue theorem to evaluate the following
integrals
16) cosa 30 de 5- 4 cos 20 17) COSI det (x + 1)? sin 3x dx 18) sin x dx (x² + 4x+5 19)
5. (a) Show that the following improper real integral is absolutely convergent cos 2x dr I (1+?}% " (b) If CR is the semicircle of radius R in the upper half plane with centre at z = 0, show carefully that e2iz lim R00JCR (1+ z2)2 d% = 0 (c) Use residue calculus to evaluate the real integral I of part (a)
5. (a) Show that the following improper real integral is absolutely convergent cos 2x dr I (1+?}% "...
Exercise 12: Residues and real integrals (a) [6+4 points) Compute the residues for all isolated singularities of the following functions (i) f(2)== (2-) tan(2), (i) 9(2):= z2 sin () (b) (4+6+5 points) Compute (using the Residue theorem) (1) cos(72) ( d, A3 := {z € C:<3), 243 := {Z EC: | = 3}, 34, (2-1)(2 + 2)2(2-4) : 43 € C:21 <3}, po 12 To (x2 + 4)2 da, 24 2 + 4 cosat. J 5 + 4 sin(t)
roo cos(x) CO Use the Residue Theorem to compute dx -00 X21 CO
roo cos(x) CO Use the Residue Theorem to compute dx -00 X21 CO
DO
NOT use a calculator. Exact answers only, no decimals.
1. (10 pt each) Evaluate the following integrals: since) dz In(In(x b. dr c. cos(x)(sin(a)2 dz d.2tan (') dr
1. (10 pt each) Evaluate the following integrals: since) dz In(In(x b. dr c. cos(x)(sin(a)2 dz d.2tan (') dr
5. Use Cauchy's residue theorem to evaluate the following integrals along the circle 121 = 4: C 22
5. Use Cauchy's residue theorem to evaluate the following integrals along the circle 121 = 4: C 22
use residue theorem to evaluate the following
integrals
sin z 21) 20) Cosx dx (r? + 1) X 22) sin mx dx 2(x² + a²² (a > 0, b>0) 23) cos ex - cos bx -dx x?
4. Evaluate the following integrals using the Residue Theorem. Justify your calculations, show the work. (10 points each) a) 12 cos a + 13 2 da b) (x2 +6x + 10)2 x sin 2x 24 13 da: c)
4. Evaluate the following integrals using the Residue Theorem. Justify your calculations, show the work. (10 points each) a) 12 cos a + 13 2 da b) (x2 +6x + 10)2 x sin 2x 24 13 da: c)