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求不定积分∫E^xsin2xDx

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∫ (e^x)sin²x dx = (1/2)∫ (e^x)(1 - cos2x) dx = (1/2)∫ e^x dx - (1/2)∫ (e^x)cos2x dx = (1/2)e^x - (1/2) • I I = ∫ (e^x)cos2x = (1/2)∫ e^x d(sin2x) = (1/2)(e^x)sin2x - (1/2)∫ (e^x)sin2x dx = (1/2)(e^x)sin2x - (1/2)(-1/...

解: ∫e^x·sin2xdx =e^x·sin2x-2∫e^xcos2xdx =e^x·sin2x-2[e^x·cos2x+2∫e^x·sin2x]dx =e^x·sin2x-2e^x·cos2x-4∫e^x·sin2x dx 得5∫e^x·sin2xdx=e^x·sin2x-2e^x·cos2x+C1 故∫e^x·sin2xdx=1/5·e^x·(sin2x-2cos2x)+C

(1)1/4*(-1+2*x)*exp(2*x)+C (2)1/4*sin(2*x)-1/2*x*cos(2*x)+C (3)-1/2*exp(-x^2)+C 前两道分部积分,最后一道换元或直接看出来

∫cos2xdx =∫cos2xd(1/2)2x =-1/2sin2x+C

e^x=sint ∫2e^x√(1-e^(2x))dx=2∫√(1-e^(2x))de^x =2∫√(1-sin²t)dsint=2∫costdsint = 2∫cos²tdt=∫1+cos2tdt =t+sin2t/2+C=t+sint√(1-sin²t)+C =t+sin2t/2+C=arctane^x+e^x√(1-e^(2x))+C

∫sin3xe^(-2x)dx=-1/3∫e^(-2x)dcos3x =-1/3cos3xe^(-2x)+1/3∫cos3xde^(-2x) =-1/3cos3xe^(-2x)-2/3∫cos3xe^(-2x)dx =-1/3cos3xe^(-2x)-2/9∫e^(-2x)dsin3x =-1/3cos3xe^(-2x)-2/9sin3xe^(-2x)+2/9∫sin3xde^(-2x) =-1/3cos3xe^(-2x)-2/9sin3xe^(-2...

这个不定积分是积不出来的!也就是,积分虽然存在,但无法用初等函数表示。

I = ∫(sinx)^2dx/e^x = (1/2)∫(1-cos2x)e^(-x)dx = (1/2)∫e^(-x)dx - (1/2)∫cos2x*e^(-x)dx = -(1/2)e^(-x) -(1/2)J, 其中 J =∫cos2x*e^(-x)dx = -∫cos2xde^(-x) = -e^(-x)cos2x - 2∫sin2xe^(-x)dx = -e^(-x)cos2x + 2∫sin2xde^(-x) = -e^(-x)co...

1、先求∫e^x*cos2x dx ∫e^x*cos2x dx = (1/2)∫e^x d(sin2x) = (1/2)(e^x)(sin2x) - (1/2)∫e^x*sin2x dx = (1/2)(e^x)(sin2x) - (1/2)(-1/2)∫e^x d(cos2x) = (1/2)(e^x)(sin2x) + (1/4)(e^x)(cos2x) - (1/4)∫e^x*cos2x dx,将最后那个积分移到左...

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