1) Evaluate the following real integrals using Residues, L***2+1&P +27 +2) (1) + drva
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
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)
Use Trip to evaluate the following integrals using Trig Sub
2. Use partial fractions to evaluate the following integrals dar 212 +12 3-4 dr:
Evaluate the following integrals
Evaluate the following integrals: 3 1 32/2 7 (i) dx 6°2712 dx x²x² + 4 (4x² +9312
4. Using the properties of the Fourier transform, evaluate the following integrals in terms of a and/or 6, where a is positive. You have to express real-valued evaluations. eat sinc(t)dt -nt - atcos (Bt)dt (10 pts) (a) (10 pts) (b) e 0 0 d [In (x)] dx Hint: teldt -1디미플
4. Using the properties of the Fourier transform, evaluate the following integrals in terms of a and/or 6, where a is positive. You have to express real-valued evaluations. eat sinc(t)dt...
ex4.19
Ex. 4.19. Use contour integration to evaluate the following real integrals dar da (a) (b) r22x5 (x21)2 da da (c) (d) (221)(a2+4)
Ex. 4.19. Use contour integration to evaluate the following real integrals dar da (a) (b) r22x5 (x21)2 da da (c) (d) (221)(a2+4)
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Evaluate the integrals of the functions graphed by using the formulas for areas of triangles and circles and by subtracting the areas below the x-axis. y 4-x-101 4 3 2 1-x-11 7 10 11 12 13 14" -1 -21 -x+8x- x-12 257 8 + 17 X Use the average of the left-endpoint (L) and right-endpoint (R) approximations to compute the integrals of the piecewise linear functions with graphs that pass through the given list of points over the...
Evaluate the following integral using residues: cos(bx)-cos(ax) I = dx. x2 Let a and b: real constants such that a > b >0. Note: cos(bz)-cos(az) has a singularity at z = 0 is removable, z2 ejbz-ejaz has a pole at the origin. Make sure to handle this point correctly 22
Using trigonometric identities and substitution, evaluate the following integrals: 1. S sin5 x dx 2. Shamrzdx 3. S V4 – 9x2dx 4. Sýsin? x dx 5. S 1 dx x2 +81
6 (5 pts cach) Evaluate the following integrals: V LLLoyzadzirdy L/ /** + udzdydis Solt LLM (2*2 + +z+z+ydzdardy 1-22 /2-22_y² wydzdyda (42+y²