1.已知∫(+∞,-∞) e^(-x^2)dx=√π,则∫(+∞,0 )e^(-ax^2)dx=2.设f(x)具有一阶连续导数,且f(0)=0,f'(x)≠0,则limx→0 ∫(x^2,0)f(t)dt/[x^2∫(x,0)f(t)dt]=
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![1.已知∫(+∞,-∞) e^(-x^2)dx=√π,则∫(+∞,0 )e^(-ax^2)dx=2.设f(x)具有一阶连续导数,且f(0)=0,f'(x)≠0,则limx→0 ∫(x^2,0)f(t)dt/[x^2∫(x,0)f(t)dt]=](/uploads/image/z/2667189-21-9.jpg?t=1.%E5%B7%B2%E7%9F%A5%E2%88%AB%EF%BC%88%2B%E2%88%9E%2C-%E2%88%9E%EF%BC%89+e%5E%EF%BC%88-x%5E2%EF%BC%89dx%3D%E2%88%9A%CF%80%2C%E5%88%99%E2%88%AB%EF%BC%88%2B%E2%88%9E%2C0+%EF%BC%89e%5E%EF%BC%88-ax%5E2%EF%BC%89dx%3D2.%E8%AE%BEf%28x%29%E5%85%B7%E6%9C%89%E4%B8%80%E9%98%B6%E8%BF%9E%E7%BB%AD%E5%AF%BC%E6%95%B0%2C%E4%B8%94f%280%29%3D0%2Cf%27%28x%29%E2%89%A00%2C%E5%88%99limx%E2%86%920+%E2%88%AB%28x%5E2%2C0%29f%28t%29dt%2F%5Bx%5E2%E2%88%AB%28x%2C0%29f%28t%29dt%5D%3D)
1.已知∫(+∞,-∞) e^(-x^2)dx=√π,则∫(+∞,0 )e^(-ax^2)dx=2.设f(x)具有一阶连续导数,且f(0)=0,f'(x)≠0,则limx→0 ∫(x^2,0)f(t)dt/[x^2∫(x,0)f(t)dt]=
1.已知∫(+∞,-∞) e^(-x^2)dx=√π,则∫(+∞,0 )e^(-ax^2)dx=
2.设f(x)具有一阶连续导数,且f(0)=0,f'(x)≠0,则limx→0 ∫(x^2,0)f(t)dt/[x^2∫(x,0)f(t)dt]=
1.已知∫(+∞,-∞) e^(-x^2)dx=√π,则∫(+∞,0 )e^(-ax^2)dx=2.设f(x)具有一阶连续导数,且f(0)=0,f'(x)≠0,则limx→0 ∫(x^2,0)f(t)dt/[x^2∫(x,0)f(t)dt]=
∫(+∞,-∞) e^(-x^2)dx=√π,
则∫(+∞,0 )e^(-ax^2)dx
=1/√a∫(+∞,0 )e^(-ax^2)d(√ax)
=√π/√a
lim(x→0) ∫(x^2,0)f(t)dt/[x^2∫(x,0)f(t)dt]
=lim(x→0) 2xf(x^2)/[2x∫(x,0)f(t)dt+x^2f(x)]
=lim(x→0) 2f(x^2)/[2∫(x,0)f(t)dt+xf(x)]
=lim(x→0) 4xf'(x^2)/[2f(x)+f(x)+xf'(x)]
=lim(x→0) 4f'(x^2)/[3f(x)/x+f'(x)]
=4f'(x^2)*lim(x→0) 1/[lim(x→0) 3f(x)/x+f'(x)]
=lim(x→0) 4f'(x^2)/(4f'(x))
=1