两道定积分证明
问题描述:
两道定积分证明
(1)∫(a,0)f(x)dx=∫(a,0)f(a-x)dx
(2)∫(π/2,0)f(sinx)dx=∫(π/2,0)f(cosx)dx
求过程
答
(1)∫(a,0)f(x)dx=∫(a,0)f(a-x)dx
t=a-x,x=a-t,dx=-dt,
x--->0时,t--->a,
x--->a时,t--->0
∫(a,0)f(a-x)dx=S(0,a)f(t)(-dt)=-S(0,a)f(t)dt=S(a,0)f(t)dt=S(a,0)f(x)dx
(2)∫(π/2,0)f(sinx)dx=∫(π/2,0)f(cosx)dx
t=π/2-x,x=π/2-t,dx=-dt
x--->0时,t--->π/2
x--->π/2时,t--->0
∫(π/2,0)f(cosx)dx
=S(0,π/2)f(cos(π/2-t))(-dt)
=-S(0,π/2)f(sint)dt
=S(π/2,0)f(sint)dt
=S(π/2,0)f(sinx)dx