Question

I'm having trouble integrating the following:

S(e^(-x)*sin(4x)dx

Whenever I try to integrate by parts I end up in this infinite loop situation. How do I get around this?

Answer 1

Is the answer that on the image

Answer 2

If it is, i will explain what i did

Answer 3

Given a complex number:
\[z=x+i y\]
Then:
\[\Im(z)=y\]
Also:
\[\huge e^{i \xi}=\cos(\xi)+i \sin(\xi)\]

\[\int\limits e^{-x}\sin(4x)dx= \int\limits \Im(e^{-x}e^{4ix})dx=\Im \left[ \int\limits e^{x(4i-1)}dx \right]\]
\[=\Im \left[ \frac{e^{x(4i-1)}}{4i-1} \right]=\Im \left[ \frac{e^{-x}(\cos(4x)+i \sin(4x))}{4i-1}*\frac{-4i- 1}{-4i-1}\right]=\]
\[e^{-x}\Im \left[ \frac{i (-4 \cos(4x)-\sin(4x))+4\sin(4x)- \cos(4x)}{17}\right]=\]
\[\frac{- e^{-x}}{17}\sin(4x)- \frac{4 e^{-x}}{17}\cos(4x)+C\]