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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/...

∫cos2xdx =∫cos2xd(1/2)2x =-1/2sin2x+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...

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

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...

An ounce of prevention

∫e^x(sinx)^2dx =1/2∫e^x(1-cos2x)dx =1/2∫(e^x-e^xcos2x)dx =1/2∫e^xdx-1/2∫e^xcos2xdx =1/2e^x-1/2∫e^xcos2xdx ∫e^xcos2xdx =∫cos2xde^x =e^xcos2x+2∫e^xsin2xdx =e^xcos2x+2∫sin2xde^x =e^xcos2x+2e^xsin2x-4∫e^xcos2xdx 5∫e^xcos2xdx=e^xcos...

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