What is the integral of [e^x *(x+1)/(x+2)^2] ?

1 Answer
Øko
Feb 21, 2018

#I=e^x/(x+2)+C#

Explanation:

We want to solve

#I=int(e^x(x+1))/(x+2)^2dx#

I don't think there is any basic technic of integration,
which solves this easily, and these types of integral,
often involves the exponential integral

However notice by the quotient rule in general we have

#((ae^x)/(x+b))'=(ae^x(x+b)-ae^x)/(x+b)^2=(ae^x(x+b-1)) /(x+b)^2#

Which leads to

#int(ae^x(x+b-1))/(x+b)^2dx=(ae^x)/(x+b)+C#

In your integral we have #a=1# and #b=2#

#I=int(e^x(x+1))/(x+2)^2dx=e^x/(x+2)+C#

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