find-0-xsin-2x-x-2-4-3-dx- Tinku Tara June 3, 2023 Integration 0 Comments FacebookTweetPin Question Number 133119 by abdomsup last updated on 19/Feb/21 find∫0∞xsin(2x)(x2+4)3dx Answered by mindispower last updated on 19/Feb/21 letf(a)=∫0∞xsin(2x)x2+adx,∀a∈R+−{0}f(a)=12Im∫−∞∞xe2ixx2+adx…..E∫−∞∞xe2ixx2+a=2iπRes(xe2ixx2+a,x=ia)=2iπ.ia.e−2a2ia=iπe−2aE⇒f(a)=π2e−2af′(a)=∫0∞−xsin(2x)(x2+a)2dxf″(a)=2∫xsin(2x)dx(x2+a)312f″(4)=∫0∞xsin(2x)(x2+4)2dx…ourintegralf′(a)=−π2ae−2a,f″(a)=π2ae−2a+π4aae−2af″(4)=5π32e−4 Answered by mathmax by abdo last updated on 20/Feb/21 Φ=∫0∞xsin(2x)(x2+4)3dx⇒Φ=x=2t∫0∞2tsin(4t)64(t2+1)3(2dt)=116∫0∞tsin(4t)(t2+1)3dt=132∫−∞+∞tsin(4t)(t2+1)3dt=132Im(∫−∞+∞te4it(t2+1)3dt)φ(z)=ze4iz(z2+1)3⇒φ(z)=ze4iz(z−i)3(z+i)3∫Rφ(z)dz=2iπRes(φ,i)andRes(φ,i)=limz→i1(3−1)!{(z−i)3φ(z)}(2)=12limz→i{ze4iz(z+i)3}(2)⇒2Res(φ,i)=limz→i{(e4iz+4ize4iz)(z+i)3−3(z+i)2ze4iz(z+i)6}(1)limz→i{(1+4iz)e4iz(z+i)−3ze4iz(z+i)4}(1)=limz→i{(1+4iz)(z+i)−3z)e4iz(z+i)4}(1)=limz→i{(i+4iz2−3z)e4iz(z+i)4}(1)=limz→i(8iz−3)e4iz+4i(4iz2−3z+1)e4iz)(z+i)4−4(z+i)3(4iz2−3z+i)e4iz(z+i)8….resttofinishthecalculus… Terms of Service Privacy Policy Contact: info@tinkutara.com FacebookTweetPin Post navigation Previous Previous post: an-object-is-placed-5cm-away-from-a-plane-mirror-the-mirror-is-rotated-tbrough-angle-20-0-about-the-point-of-incidence-calculate-the-shortest-distance-between-the-old-and-the-new-position-of-the-imNext Next post: 1-sin-3-x-dx- Leave a Reply Cancel replyYour email address will not be published. Required fields are marked *Comment * Name * Save my name, email, and website in this browser for the next time I comment.