二重积分计算：∫[0,a]dx∫[0,x] f ´(y)/√[(a-x)(x-y)] dy

题目

二重积分计算：∫[0,a]dx∫[0,x] f ´(y)/√[(a-x)(x-y)] dy

答案

原式=∫f'(y)dy∫dx/√[(a-x)(x-y)] (交换积分顺序)
=2∫f'(y)dy∫dt/(t²+1) (设√[(x-y)/(a-x)]=t,当x=y时,t=0.当x=a时,t=+∞.再化简)
=π∫f'(y)dy (∫dt/(t²+1)=π/2)
=π[f(a)-f(0)].