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
To determine which answer is correct, let's compare the two solutions and simplify them:
Solution A: e^x*sin(3x) + 3e^x*cos(3x)
Solution B: (1/10)e^x(sin(3x) - 3cos(3x))
To simplify Solution A, we can use the trigonometric identity: cos(a)sin(b) - sin(a)cos(b) = sin(a - b).
Solution A: e^x*sin(3x) + 3e^x*cos(3x)
= e^x*(sin(3x) + 3cos(3x))
= e^x*[sin(3x) - (-3cos(3x))]
= e^x*[sin(3x) - 3cos(3x)]
Comparing Solution A with Solution B, we can see that they are the same up to a constant factor of (1/10). Therefore, both answers are correct, taking into account that the constant of integration (C) may vary between different sources or textbooks.