Integral of e^x(sinx+cosx)

In this tutorial we shall find some different type of function like 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 a formula for integration in which exponential function into some function plus its derivative, we have

\int  {{e^x}\left[ {f\left( x \right) + f'\left( x \right)} \right]dx = {e^x}f\left(  x \right) + c}


Here we have given function f\left( x \right) = \sin x, 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}

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