Integration of e^x Sin x

In this tutorial we shall drive the integral of e^x into sine function, and this integral can be evaluate by using integration by parts method.

The integration of the form

I =  \int {{e^x}\sin xdx} \,\,\,\,{\text{ -  -  - }}\left( {\text{i}} \right)


Here first function is f\left(  x \right) = {e^x} and second function will be g\left( x \right) = \sin x
By using integration by parts formula

\int  {\left[ {f\left( x \right)g\left( x \right)} \right]dx = f\left( x \right)\int  {g\left( x \right)dx - \left[ {\frac{d}{{dx}}f\left( x \right)\left( {\int  {g\left( x \right)} dx} \right)} \right]} dx}


Equation (i) will becomes

\begin{gathered} I = {e^x}\int {\sin xdx - \int {\left[  {\frac{d}{{dx}}{e^x}\left( {\int {\sin xdx} } \right)} \right]} dx} \\ \Rightarrow I = {e^x}\left( { - \cos x}  \right) - \int {\left[ {{e^x}\left( { - \cos x} \right)} \right]} dx \\ \Rightarrow I =  - \cos x{e^x} + \int {{e^x}\cos x} dx \\ \end{gathered}


Again using integration by parts formula, we have

\begin{gathered} I =  -  \cos x{e^x} + {e^x}\int {\cos x} dx - \int {\left[ {\frac{d}{{dx}}{e^x}\left(  {\int {\cos x} } \right)dx} \right]} dx \\ \Rightarrow I =  - \cos x{e^x} + {e^x}\sin x - \int {{e^x}\sin  x} dx \\ \end{gathered}


But using I = \int  {{e^x}\sin xdx} , we have

\begin{gathered} I =  -  {e^x}\cos x + {e^x}\sin x - I \\ \Rightarrow I + I =  - {e^x}\cos x + {e^x}\sin x \\ \Rightarrow 2I =  - {e^x}\cos x + {e^x}\sin x \\ \Rightarrow I = \frac{1}{2}{e^x}\left( {\sin  x - \cos x} \right) + c \\ \Rightarrow \int {{e^x}\sin x} dx =  \frac{1}{2}{e^x}\left( {\sin x - \cos x} \right) + c \\ \end{gathered}