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sin-2tan-1-x-x-x-the-limit-0-




Question Number 127885 by psyche last updated on 02/Jan/21
∫(((sin (2tan^(−1) (x)+x))/x))  the limit [0,∞)
(sin(2tan1(x)+x)x)thelimit[0,)
Answered by Lordose last updated on 02/Jan/21
  Ω = ∫_0 ^( ∞) ((sin(2tan^(−1) (x)+x))/x)dx  sin(2tan^(−1) (x)+x)=sin(2tan^(−1) (x))cos(x)+cos(2tan^(−1) (x))sin(x)    N.B::  (((sin(2tan^(−1) (x))= ((2x)/(1+x^2 )))),((cos(2tan^(−1) (x))= ((1−x^2 )/(1+x^2 )) )) )  Ω = ∫_0 ^( ∞) (((1/(1+x^2 ))(2xcos(x)+(1−x^2 )sin(x)))/x)  Ω = ∫_0 ^( ∞) ((2cos(x))/(1+x^2 ))dx + ∫_0 ^( ∞) ((sin(x))/(x(1+x^2 )))dx − ∫_0 ^( ∞) ((xsin(x))/(1+x^2 ))dx  Ω = 2∙(π/(2e)) + ∫_0 ^( ∞) ((sin(x))/(x(1+x^2 )))dx − ∫_0 ^( ∞) ((sin(x))/x)dx + ∫_0 ^( ∞) ((sin(x))/(x(1+x^2 )))dx  Ω = (π/e) − (π/2) + 2Φ   Φ = ∫_0 ^( ∞) ((sin(x))/(x(1+x^2 )))dx = (π/2)(1−e^(−1) )  Ω = (π/e) − (π/2) + π − (π/e) = (π/2)  ★L𝛗rD ∅sE
Ω=0sin(2tan1(x)+x)xdxsin(2tan1(x)+x)=sin(2tan1(x))cos(x)+cos(2tan1(x))sin(x)N.B::(sin(2tan1(x))=2x1+x2cos(2tan1(x))=1x21+x2)Ω=011+x2(2xcos(x)+(1x2)sin(x))xΩ=02cos(x)1+x2dx+0sin(x)x(1+x2)dx0xsin(x)1+x2dxΩ=2π2e+0sin(x)x(1+x2)dx0sin(x)xdx+0sin(x)x(1+x2)dxΩ=πeπ2+2ΦΦ=0sin(x)x(1+x2)dx=π2(1e1)Ω=πeπ2+ππe=π2\boldsymbolLϕrD\boldsymbolsE

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