Integral of e^x(ln x+1/x)

In this tutorial we shall find some different type of function like integral of e^x(ln x+1/x).

The integral of e^x(ln x+1/x) is of the form

I =  \int {{e^x}\left( {\ln x + \frac{1}{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) = \ln x, differentiate with respect to variable x we have f'\left( x  \right) = \frac{1}{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( {\ln x} \right) + c \\ \Rightarrow \int {{e^x}\left( {\ln x +  \frac{1}{x}} \right)dx}  = {e^x}\ln x + c \\ \end{gathered}