计算积分I=∫+∞0e-ax-e-bxxcosmxdx，a，b均为常数．

Question

计算积分 I=

∫

+∞ 0

(e-ax-e-bx)/(x)cosmxdx，a，b均为常数．

Answer 1

对∀y∈[a，b]，|e^-yxcosmx|≤e^-ax．而

∫

+∞ 0

e-axdx收敛，故

∫

+∞ 0

e-yxcosmxdx在[a，b]一致收敛．
又被积函数f（x，y）=e^-xycosmx在0≤x≤+∞，a≤y≤b上连续，
故交换积分顺序，积分I得值不变．
于是
I=∫

+∞

0

(e-ax-e-bx)/(x)cosmxdx=

∫

+∞ 0

dx

∫

b a

e-xycosmxdy
=

∫

b a

dy

∫

+∞ 0

e-xycosmxdx=

∫

b a

(1)/(1+y2)dy
=arctanb-arctana