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

The integral of ((e^x+1)^2)/(e^x) is e^x+2x-1/(e^x)+C

integral of ((e^x+1)^2)/(e^x)

\int \frac{( e ^{x} + 1 ) ^{2}}{e ^{x}} d x

e^{x} + 2 x - \frac{1}{e ^{x}} + C

Solution steps

\int \frac{( e ^{x} + 1 ) ^{2}}{e ^{x}} d x

Expand

\frac{( e ^{x} + 1 ) ^{2}}{e ^{x}} : e^{x} + 2 + \frac{1}{e ^{x}}

= \int e^{x} + 2 + \frac{1}{e ^{x}} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int e^{x} d x + \int 2 d x + \int \frac{1}{e ^{x}} d x

\int e^{x} d x = e^{x}

\int 2 d x = 2 x

\int \frac{1}{e ^{x}} d x = - \frac{1}{e ^{x}}

= e^{x} + 2 x - \frac{1}{e ^{x}}

Add a constant to the solution

= e^{x} + 2 x - \frac{1}{e ^{x}} + C

\frac{d}{d x} (\frac{1}{27} (2 x - 1)^{3} (6 - 3 x) + 1)

\int_{0}^{\frac{π}{4}} sin (4 x) d x

in v erse l a pl a ce \frac{- 3}{( s - 1 ) ^{2}}

d er i v a t i v e f (x) = \frac{x ^{3}}{3} + 4 x^{2} + 12 x + 19

e^{x} (y - 1) d x + 2 (e^{x} + 4) d y = 0

What is the integral of ((e^x+1)^2)/(e^x) ?
The integral of ((e^x+1)^2)/(e^x) is e^x+2x-1/(e^x)+C