谢谢
追答前面打错了,更正一下
sin(tx)/x=∫(0→t)cos(xy)dy
所以,
∫(0→+∞)e^(-x)·sin(tx)/x·dx
=∫(0→+∞)e^(-x)·[∫(0→t)cos(xy)dy]·dx
=∫(0→t)dy∫(0→+∞)e^(-x)·cos(xy)·dx
=∫(0→t)1/(1+y²)dy
=arctany |(0→t)
=arctant
【附注】内层积分的计算
∫(0→+∞)e^(-x)·cos(xy)·dx
=-∫(0→+∞)cos(xy)·d[e^(-x)]
=-e^(-x)cos(xy) |(0→+∞)
-∫(0→+∞)e^(-x)·ysin(xy)·dx
=1-∫(0→+∞)e^(-x)·ysin(xy)·dx
=1+∫(0→+∞)ysin(xy)·d[e^(-x)]
=1+e^(-x)·ysin(xy) |(0→+∞)
-∫(0→+∞)e^(-x)·y²cos(xy)·dx
=1-y²∫(0→+∞)e^(-x)·cos(xy)·dx
移项可得:
(1+y²)∫(0→+∞)e^(-x)·cos(xy)·dx=1
∴∫(0→+∞)e^(-x)·cos(xy)dx=1/(1+y²)