e^xsinx^2dx不定积分
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I = ∫e^x(sinx)^2dx = (1/2)∫e^x(1-cos2x)dx
= (1/2)e^x - (1/2)∫e^xcos2xdx
其中 J = ∫e^xcos2xdx = ∫cos2xde^x
= e^xcos2x + 2∫sin2xe^xdx
= e^xcos2x + 2e^xsin2x - 2∫cos2xe^xdx
= e^x(cos2x + 2sin2x) - 2J,
则 J = (1/3)e^x(cos2x + 2sin2x),
I = (1/2)e^x - (1/6)e^x(cos2x + 2sin2x) + C
= (1/6)e^x(3-cos2x - 2sin2x) + C
= (1/2)e^x - (1/2)∫e^xcos2xdx
其中 J = ∫e^xcos2xdx = ∫cos2xde^x
= e^xcos2x + 2∫sin2xe^xdx
= e^xcos2x + 2e^xsin2x - 2∫cos2xe^xdx
= e^x(cos2x + 2sin2x) - 2J,
则 J = (1/3)e^x(cos2x + 2sin2x),
I = (1/2)e^x - (1/6)e^x(cos2x + 2sin2x) + C
= (1/6)e^x(3-cos2x - 2sin2x) + C
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