1、
∫x³e^(-x²) dx,t=-x²,dt=-2x dx,dx=(-1/(2x))dt
原式= (-1/2)∫(-t)e^t dt
= (1/2)∫t de^t,分部积分法
= (1/2)t*e^t - (1/2)∫e^t dt,分部积分法
= (1/2)t*e^t - (1/2)e^t + C
= (1/2)(t-1)e^t + C
= (-1/2)(x²+1)e^(-x²) + C
2、看不明白你写的什么,f(2x)的导数f(x)的原函数为sinx / x即∫f(x) dx = sinx / xf(x) = (sinx / x)' = (xcosx-sinx)/x^2f'(x) = [(2-x^2)sinx-2xcosx]/x^3f'(2x) = {[2-(2x)^2]sin(2x)-2(2x)cos2x}/(2x)^3 = [(2-4x^2)sin2x-4xcos2x]/(8x^3)= [(1-2x^2)sin2x-2xcos2x]/(4x^3)∫x*f'(2x) dx= ∫[(1-2x^2)sin2x-2xcos2x]/(4x^2) dx= (1/4)∫(sin2x-2x^2*sin2x-2xcos2x)/x^2 dx= (1/4)∫(sin2x)/x^2 dx - (1/2)∫(cos2x)/x dx - (1/2)∫sin2x dx= (1/4)∫(sin2x)/x^2 dx - (1/4)∫(1/x) dsin2x - (1/4)∫sin2x d(2x)，分部积分法= (1/4)∫(sin2x)/x^2 dx - (1/4)[(sin2x)/x - ∫sin2x d(1/x)] - (1/4)(-cos2x)，分部积分法= (1/4)∫(sin2x)/x^2 dx - (sin2x)/(4x) - (1/4)∫(sin2x)/x^2 dx + (1/4)cos2x= (1/4)[cos2x - (sin2x)/x] + C