Question

请问这题该怎么做，求完整步骤

高数 变限积分题，用分部积分法求解

Answer 1

f(x)=∫(1->x^2) e^(-t^2) dt
f'(x) = 2x . e^(-x^4)
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∫(0->1) xf(x) dx
=(1/2)∫(0->1) f(x) dx^2
=(1/2) [ x^2. f(x) ]|(0->1) -(1/2)∫(0->1) x^2.f'(x) dx
=-(1/2)∫(0->1) x^2.f'(x) dx
=-(1/2)∫(0->1) x^2.[2x . e^(-x^4)] dx
=-∫(0->1) x^3 . e^(-x^4) dx
=(1/4) ∫(0->1) e^(-x^4) d(-x^4)
=(1/4) [ e^(-x^4) ]|(0->1)
=(1/4) [ e^(-1) - 1]