How do you integrate $\int \frac{e^{2 x} + 2 e^{x} + 1}{e^{x}} d x$ $int (e^(2x)+2e^x+1)/(e^x)dx$ ?

Question

How do you integrate $\int \frac{e^{2 x} + 2 e^{x} + 1}{e^{x}} d x$ $int (e^(2x)+2e^x+1)/(e^x)dx$ ?

1 Answer

Impact of this question

14449 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-11-03T06:31:24

$= e^{x} + 2 x - e^{- x} + C$ $=e^x+2x-e^(-x)+C$

Explanation:

Inegrating the given rational function is determined by decomposing
$" "$
the given fraction into partial ones
$" "$
$\int \frac{e^{2 x} + 2 e^{x} + 1}{e^{x}} d x$ $int(e^(2x)+2e^x+1)/e^xdx$
$" "$
$= \int \frac{e^{2 x}}{e^{x}} d x + \int \frac{2 e^{x}}{e^{x}} d x + \int \frac{1}{e^{x}} d x$ $=int(e^(2x))/e^xdx +int(2e^x)/e^xdx+int1/e^xdx$
$" "$
$= \int e^{x} d x + \int 2 d x + \int e^{- x} d x$ $=inte^xdx+int2dx+inte^(-x)dx$
$" "$
Knowing that
$" "$
$d (e^{x}) = e^{x} d x$ $color(red)(d(e^x)=e^xdx$ and
$" "$
$d (e^{- x}) = - e^{x} \Rightarrow e^{- x} = - d (e^{- x})$ $color(purple)(d(e^(-x))=-e^xrArre^(-x)=-d(e^(-x))$
$" "$
$= \int d (e^{x}) + \int 2 d x + \int - d (e^{- x})$ $=intcolor(red)(d(e^x))+int2dx+intcolor(purple)(-d(e^(-x))$
$" "$
$= e^{x} + 2 x - e^{- x} + C$ $=e^x+2x-e^(-x)+C$

Science

Math

Humanities

... and beyond

How do you integrate $\int \frac{e^{2 x} + 2 e^{x} + 1}{e^{x}} d x$ $int (e^(2x)+2e^x+1)/(e^x)dx$ ?

1 Answer

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

How do you integrate ∫e2x+2ex+1exdxint (e^(2x)+2e^x+1)/(e^x)dx?

1 Answer

Explanation:

Related questions

Impact of this question

How do you integrate $\int \frac{e^{2 x} + 2 e^{x} + 1}{e^{x}} d x$ $int (e^(2x)+2e^x+1)/(e^x)dx$ ?