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

The integral of e^{-x}ln(1+e^x) is -e^{-x}ln(1+e^x)-ln|1+e^x|+x+C

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

\int e^{- x} ln (1 + e^{x}) d x

- e^{- x} ln (1 + e^{x}) - ln ∣ 1 + e^{x} ∣ + x + C

Solution steps

\int e^{- x} ln (1 + e^{x}) d x

Apply Integration By Parts

: - e^{- x} ln (1 + e^{x}) - \int - \frac{1}{1 + e ^{x}} d x

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

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

= - e^{- x} ln (1 + e^{x}) - (ln ∣ 1 + e^{x} ∣ - x)

Simplify

= - e^{- x} ln (1 + e^{x}) - ln ∣ 1 + e^{x} ∣ + x

Add a constant to the solution

= - e^{- x} ln (1 + e^{x}) - ln ∣ 1 + e^{x} ∣ + x + C

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What is the integral of e^{-x}ln(1+e^x) ?
The integral of e^{-x}ln(1+e^x) is -e^{-x}ln(1+e^x)-ln|1+e^x|+x+C