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evaluate-sin-x-x-dx-




Question Number 243 by 123456 last updated on 25/Jan/15
evaluate  ∫_(−∞) ^(+∞) ((sin x)/x)dx
evaluate+sinxxdx
Answered by prakash jain last updated on 17/Dec/14
Let us consider ∫_0 ^∞  ∫_0 ^∞  e^(−st) sin t dt ds  ∫_0 ^∞  ∫_0 ^∞  e^(−st) sin t dt ds=∫_0 ^∞  (1/(1+s^2 ))ds=[arctan s]_0 ^∞ =(π/2)...(1)  ∫_0 ^∞  ∫_0 ^∞  e^(−st) sin t dt ds=∫_0 ^∞  ∫_0 ^∞  e^(−st) sin t ds dt  =∫_0 ^∞  ((sin t)/t) dt....(2)  From (1) and (2)  ∫_0 ^∞  ((sin t)/t) dt =(π/2)  ∫_(−∞) ^∞  ((sin t)/t) dt=2∫_0 ^∞  ((sin t)/t) dt=π
Letusconsider00estsintdtds00estsintdtds=011+s2ds=[arctans]0=π2(1)00estsintdtds=00estsintdsdt=0sinttdt.(2)From(1)and(2)0sinttdt=π2sinttdt=20sinttdt=π