Question

高数积分题，请帮帮忙，谢谢！题目如图。

Answer 1

let
x= asiny
dx= acosy dy
x=0 , y=0
x=a ,y=π/2

∫(0->a) dx/(x+√(a^2-x^2)
=∫(0->π/2) cosy /(siny+cosy ) dy

let
cosy = k1(siny +cosy) + k2(cosy -siny)
k1-k2 =0 (1)
k1+k2=1 (2)
(1)+(2)
k1=1/2

from (1)
k2 =1/2

∫(0->a) dx/(x+√(a^2-x^2)
=∫(0->π/2) cosy /(siny+cosy ) dy
=(1/2)∫(0->π/2) (siny+cosy) /(siny+cosy ) dy
+(1/2)∫(0->π/2) (cosy-siny) /(siny+cosy ) dy
=(1/2) [y]|(0->π/2) + (1/2) [ln|siny+cosy|] |(0->π/2)
=π/4

追问

谢谢 可以用笔写拍照吗，有点头绪，但还不太明白

请问下最后一步是怎么算了等于π/4的

追答

(1/2) [y]|(0->π/2) =π/4

(1/2) [ln|siny+cosy|] |(0->π/2) =0

追问

请问这个怎么做