Question

\[\large \int\limits \frac{ e ^{2x}+2e^x+1 }{ e^x }dx\]
I think I just need help with figuring out the u for the u sub for this one :3

Answer 1

I did this for my second step\[\large \int\limits \frac{ (e^x+1)^2 }{ e^x }dx\]

Answer 2

or should I try to separate it?

Answer 3

Hmm, I was thinking \(u=e^x\) would be nice.

Answer 4

Alternatively, this doesn't really need a \(u\) sub.

Answer 5

or something like this?
\[\int\limits \frac{ e^x(e^x+2) }{ e^x }dx + \int\limits \frac{1}{e^x}dx\]

Answer 6

\[\begin{split}

\int\limits \frac{ e ^{2x}+2e^x+1 }{ e^x }dx
&= \int\limits \frac{e ^{2x}}{e^x}+\frac{2e^x}{e^x}+\frac{1 }{ e^x }dx \\
&= \int\limits e^x+2+e^{-x}~dx \\
&= \int\limits e^x~dx+\int\limits2~dx+\int\limits e^{-x}~dx \\
\end{split}
\]

Answer 7

ohhh I see :) thanks!

Answer 8

so it would be e^x + 2x - e^(-x) + C ?

Answer 9

Sure.

Answer 10

As a note \(e^x-e^{-x} = 2\sinh(x)\). Though it probably doesn't matter here.

Answer 11

yeah, it doesn't, but thanks! :D