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

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

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

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

2 x + e^{x} + C

Solution steps

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

Apply u-substitution

= \int \frac{1}{u} d u

Apply the Power Rule

= 2 u^{\frac{1}{2}}

Substitute back

u = x + e^{x}

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

Apply radical rule:

a^{\frac{1}{n}} = n a

= 2 x + e^{x}

Add a constant to the solution

= 2 x + e^{x} + C

\frac{\partial}{\partial x} (x^{10} e^{x y})

x \to 2 - lim (\frac{2 x + 8}{x + 2})

\frac{d}{d x} (\frac{π x ^{2}}{4})

\int \frac{1}{4 y ^{2} + 12 y + 9} d y

f (t) = \frac{t ^{3}}{3}

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