Question

16题求不定积分，求详解

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Answer 1

分部积分:
=(1/2)∫cos3xde^2x
=(1/2)[e^2xcos3x一∫e^2xdcos3x]
=(1/2)[e^2xcos3x十∫3e^2xsin3xdx]
=e^2xcos3x/2十(3/4)∫sin3xde^2x
=
e^2xcos3x/2十(3/4)e^2xsin3x一(3/4)∫e^2xdsin3x
=
e^2xcos3x/2十(3/4)e^2xsin3x一(9/4)∫e^2xcos3xdx
注意，最后一项又是原积分，设该积分为f，则
f=
e^2xcos3x/2十(3/4)e^2xsin3x一(9/4)f
(13/4)f=
e^2xcos3x/2十(3/4)e^2xsin3x
f(x)=e^2x(2cos3x十3sin3x)/13十C