What is the integral of (ln(x))/(e^x) ?

The integral of (ln(x))/(e^x) is -e^{-x}ln(x)+\Ei(-x)+C

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integral of (ln(x))/(e^x)

\int \frac{ln ( x )}{e ^{x}} d x

- e^{- x} ln (x) + Ei (- x) + C

Solution steps

\int \frac{ln ( x )}{e ^{x}} d x

Apply exponent rule:

\frac{1}{a ^{b}} = a^{- b}

\frac{ln ( x )}{e ^{x}} = (ln (x)) e^{- x}

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

Apply Integration By Parts

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

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

\int - \frac{e ^{- x}}{x} d x = - Ei (- x)

= - e^{- x} ln (x) - (- Ei (- x))

Simplify

= - e^{- x} ln (x) + Ei (- x)

Add a constant to the solution

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

f (x) = x^{2} + 2 x + 1

n = 1 \sum \infty \frac{x ^{n}}{8 ^{n}}

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

sin (3 t)

What is the integral of (ln(x))/(e^x) ?
The integral of (ln(x))/(e^x) is -e^{-x}ln(x)+\Ei(-x)+C