Question

高数 定积分的一个问题？

第五题 谢谢

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Answer 1

Answer 2

g(x) =∫(0->x) f(t) (x-t) dt
g(0) =0
g(x)
=∫(0->x) f(t) (x-t) dt
=x∫(0->x) f(t) dt -∫(0->x) tf(t) dt
g'(x)
=∫(0->x) f(t) dt + xf(x) -xf(x)
=∫(0->x) f(t) dt
f(x)=∫(0->x) [∫(0->t) f(u) du] dt
f(0) =0
f(x)=∫(0->x) [∫(0->t) f(u) du] dt

f'(x) = ∫(0->x) f(u) du = g'(x)
g(0)=f(0)
=>
g(x) = f(x)
∫(0->x) f(t) (x-t) dt =∫(0->x) [∫(0->t) f(u) du] dt