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given (e^x)sin3x = Im((e^x)(e^i3x)) Integrate (e^x)sin3x.dx

You can just take the imaginary part of the integral of ((e^x)(e^i3x))= e^([3i +1)x] which is IM(3i +1)e^(3ix + x)
= IM (3i +1)e^x*e^(3ix)
= IM {(3i +1)*e^x* [ cos (3x) + i sin (3x)]
= e^x* sin (3x)+ 3 e^x cos (3x)

Check my algebra. I don't guarantee it

I follow your working but my book says the answer is (1/10)e^x(sin3x - 3cos3x) + C
so is the book wrong or can I manipulate your answer in some way to get the same as the book
By the way Thanks for your fast response

I follow your working but my book says the answer is (1/10)e^x(sin3x - 3cos3x) + C
so is the book wrong or can I manipulate your answer in some way to get the same as the book
By the way Thanks for your fast response

I get e^x(cos3x-3sin3x)
so now there 3 possible answers

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