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