Question

高数 请问解法二划线部分怎么来的？

Answer 1

对称区间内对奇函数积分为零，添加的那项对整体积分没影响

Answer 2

(1)
tany = 1/coty
=> arctany = arcot(1/y)
=> arctan(e^x) = arccot[ 1/e^(x) ] = arccot[e^(-x)]
(ii)
arctan(e^x) +arccot(e^x) =1
I

=5∫(-π/4-> π/4) 5cosx .arctane^x dx
=5∫(-π/4-> π/4) 5cosx .arccote^x dx
2I
=5∫(-π/4-> π/4) cosx .arctane^x dx +5∫(-π/4-> π/4) 5cosx .arccote^x dx
=5∫(-π/4-> π/4) cosx .(arctane^x +arccote^x )dx
=5∫(-π/4-> π/4) cosx dx
I = (5/2) ∫(-π/4-> π/4) cosx dx

追问

arctan(e^x) +arccot(e^x) =1 请问这一步为什么等于1

追答

不好意思应该是 arctan(e^x) +arccot(e^x) =π/2

tan(A+B)= (tanA+tanB)/(1- tanA.tanB)

A+B

= arctan[(tanA+tanB)/(1- tanA.tanB) ]

A=arctan(e^x) , B=arctan(e^(-x) )

arctan(e^x) + B=arctan(e^(-x) )

=arctan[ e^x+e^(-x) )/(1- 1) ]

=arctan（∞）

=π/2