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

The integral of ((e^x+2)e^x)/(sqrt(e^x+1)) is 2/3 (e^x+1)^{3/2}+2sqrt(e^x+1)+C

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

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

\frac{2}{3} (e^{x} + 1)^{\frac{3}{2}} + 2 e^{x} + 1 + C

Solution steps

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

Expand

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

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

\int \frac{e ^{2 x}}{e ^{x} + 1} d x = \frac{2}{3} (e^{x} + 1)^{\frac{3}{2}} - 2 e^{x} + 1

\int \frac{2 e ^{x}}{e ^{x} + 1} d x = 4 e^{x} + 1

= \frac{2}{3} (e^{x} + 1)^{\frac{3}{2}} - 2 e^{x} + 1 + 4 e^{x} + 1

Simplify

= \frac{2}{3} (e^{x} + 1)^{\frac{3}{2}} + 2 e^{x} + 1

Add a constant to the solution

= \frac{2}{3} (e^{x} + 1)^{\frac{3}{2}} + 2 e^{x} + 1 + C

What is the integral of ((e^x+2)e^x)/(sqrt(e^x+1)) ?
The integral of ((e^x+2)e^x)/(sqrt(e^x+1)) is 2/3 (e^x+1)^{3/2}+2sqrt(e^x+1)+C