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

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

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integral of (sqrt(1+e^{-x)})/(e^x)

integral

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

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

Solution steps

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

Apply u-substitution

= \int - u d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int u d u

Apply the Power Rule

= - \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 1 + e^{- x}

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

Add a constant to the solution

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

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