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EXERCISE 20.7 Evaluate the following integrals 2x de (k>1) k+cose de sin' e de .lpl<1 1-2p sin 8+p

EXERCISE 20.7 Evaluate the following integrals 2x de (k>1) k+cose de sin' e de .lpl<1 1-2p sin 8+p

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.4: Values Of The Trigonometric Functions
Problem 23E
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20.7.3 Improper Real Integrals of the types («) cos ax dx and jA) sin ax de (Fourler Integrals). We can express these integrals as cos ax dx - Re and wheme a is real and positive. Consider the contour integral f fex" dz, where C is the dosed contour given by c-CU -R, RL as shown in Fig (204). Using the residue theorem, we get or, -(2043) where the sum an the right side is over the residues of fleye" at its poles in the upper half-plane. Taylor Sarles, Laurent Series and The Residue Theorem 1167 1 Also 2niRes = 2ni. 2* 2 Hence, as R (20.85) gives de -(20.56) 1 Next, for z<0, refer to Fig (20.13b), the function f(u)= has singularity w = i inside 2n w+1 C and thus by residue theorem 1 dw = 1 do+ w +1 dw = 2ni Res +1 -(2087) -i 2x 1 =2ni 2n +1 Also 2ni Res 2* -2i On the same lines, as above đo - 0 when Rm, and hence (20.56) gives -(208) Henoe, from (20.87) and (2088) we abtain ft)=
Transcribed Image Text:20.7.3 Improper Real Integrals of the types («) cos ax dx and jA) sin ax de (Fourler Integrals). We can express these integrals as cos ax dx - Re and wheme a is real and positive. Consider the contour integral f fex" dz, where C is the dosed contour given by c-CU -R, RL as shown in Fig (204). Using the residue theorem, we get or, -(2043) where the sum an the right side is over the residues of fleye" at its poles in the upper half-plane. Taylor Sarles, Laurent Series and The Residue Theorem 1167 1 Also 2niRes = 2ni. 2* 2 Hence, as R (20.85) gives de -(20.56) 1 Next, for z<0, refer to Fig (20.13b), the function f(u)= has singularity w = i inside 2n w+1 C and thus by residue theorem 1 dw = 1 do+ w +1 dw = 2ni Res +1 -(2087) -i 2x 1 =2ni 2n +1 Also 2ni Res 2* -2i On the same lines, as above đo - 0 when Rm, and hence (20.56) gives -(208) Henoe, from (20.87) and (2088) we abtain ft)=
EXERCISE 20.7 Evaluate the following integrals 2x de de (k> 1) k+cos e e de ở 1-2p sin 6+plPl<1 2x 13- 12 cas 20 cos 20
Transcribed Image Text:EXERCISE 20.7 Evaluate the following integrals 2x de de (k> 1) k+cos e e de ở 1-2p sin 6+plPl<1 2x 13- 12 cas 20 cos 20
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