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

= \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}}

= - \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

\frac{x ^{2}}{e ^{x}}

(x + 2)^{5}

ln (x^{2} + 2)

8 x - 3

\frac{( 2 x + 5 )}{( 7 x + 22 )}

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