The Integral of e^x(sinx+cosx)

In this tutorial we shall find a different type of function known as the integral of e^x(sinx+cosx).

The integral of e^x(sinx+cosx) is of the form
\[I = \int {{e^x}\left( {\sin x + \cos x} \right)dx} \]

Using the following formula for integration
\[\int {{e^x}\left[ {f\left( x \right) + f’\left( x \right)} \right]dx = {e^x}f\left( x \right) + c} \]

Here we are given function $$f\left( x \right) = \sin x$$, and differentiate with respect to variable x we have $$f’\left( x \right) = \cos x$$

Now using the formula \[\int {{e^x}\left[ {f\left( x \right) + f’\left( x \right)} \right]dx = {e^x}f\left( x \right) + c} \]
\[\begin{gathered} I = {e^x}\left( {\sin x} \right) + c \\ \Rightarrow \int {{e^x}\left( {\sin x + \cos x} \right)dx} = {e^x}\sin x + c \\ \end{gathered} \]