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Question Number 75721 by Gazella thomsonii last updated on 15/Dec/19
solve this complex integral∫_(−∞) ^(+∞)  (e^(it) /( (√(1+t^2 ))))dt
$$\mathrm{solve}\:\mathrm{this}\:\mathrm{complex}\:\mathrm{integral}\int_{−\infty} ^{+\infty} \:\frac{{e}^{\boldsymbol{{i}}{t}} }{\:\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }}\mathrm{d}{t} \\ $$
Commented by MJS last updated on 15/Dec/19
∫_(−∞) ^∞ (e^(it) /( (√(t^2 +1))))dt=2∫_0 ^∞ ((cos t)/( (√(t^2 +1))))dt+i∫_(−∞) ^∞ ((sin t)/( (√(t^2 +1))))dt=  =2∫_0 ^∞ ((cos t)/( (√(t^2 +1))))dt  but I cannot solve this...
$$\underset{−\infty} {\overset{\infty} {\int}}\frac{\mathrm{e}^{\mathrm{i}{t}} }{\:\sqrt{{t}^{\mathrm{2}} +\mathrm{1}}}{dt}=\mathrm{2}\underset{\mathrm{0}} {\overset{\infty} {\int}}\frac{\mathrm{cos}\:{t}}{\:\sqrt{{t}^{\mathrm{2}} +\mathrm{1}}}{dt}+\mathrm{i}\underset{−\infty} {\overset{\infty} {\int}}\frac{\mathrm{sin}\:{t}}{\:\sqrt{{t}^{\mathrm{2}} +\mathrm{1}}}{dt}= \\ $$$$=\mathrm{2}\underset{\mathrm{0}} {\overset{\infty} {\int}}\frac{\mathrm{cos}\:{t}}{\:\sqrt{{t}^{\mathrm{2}} +\mathrm{1}}}{dt} \\ $$$$\mathrm{but}\:\mathrm{I}\:\mathrm{cannot}\:\mathrm{solve}\:\mathrm{this}… \\ $$

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