高中数学求助：已知函数，求导，谢谢！
最新回答
紫珺婳浅
2024-04-23 02:38:15
I(2n) = ∫(0->π/2) (cosx)^(2n) dx
I6
=∫(0->π/2) (cosx)^6 dx
=∫(0->π/2) (cosx)^5 dsinx
=[sinx.(cosx)^5]|(0->π/2) +5∫(0->π/2) (sinx)^2.(cosx)^4 dx
=5∫(0->π/2) (sinx)^2.(cosx)^4 dx
=5∫(0->π/2) [ 1-(cosx)^2].(cosx)^4 dx
6I6 = 5I4
I6 = (5/6)I4
=(5/6)(3/4)(1/2)I0
=(5/16) ∫(0->π/2) dx
=(5/32)π
ie
∫(0->π/2) (cosx)^6 dx=(5/32)π
(2)
f(x) =∫(1->√x) e^(-t^2) dt
=>f(1) = 0
两边取导
f'(x) = [1/(2√x)] e^(-x)
--------
∫(0->1) f(x)/√x dx
=2∫(0->1) f(x)d√x
=2[√x.f(x) ] |(0->1) -2∫(0->1) √x.f'(x)dx
=0 -2∫(0->1) √x.f'(x)dx
=-2∫(0->1) √x.{ [1/(2√x)] e^(-x) }dx
=-∫(0->1) e^(-x) dx
=[e^(-x)]|(0->1)
=e^(-1) -1
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