Chapter 8 - Techniques of Integration - 8.1 Integration by Parts - Exercises - Page 395: 15

$$\int e^{-x}sin\,x\,dx=-\frac{e^{-x}}{2}(cos\,x+sin\,x)+C$$

$$\int e^{-x}sin\,x\,dx=-e^{-x}cos\,x-\int e^{-x}cos\,x\,dx$$ $$=-e^{-x}cos\,x-(e^{-x}sin\,x+\int e^{-x}sin\,x\,dx)$$ $$2\int e^{-x}sin\,x\,dx=-e^{-x}cos\,x-e^{-x}sin\,x$$ $$\int e^{-x}sin\,x\,dx=-\frac{e^{-x}}{2}(cos\,x+sin\,x)+C$$