What is the integral of (1-xln(x))/(xe^x) ?

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

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integral of (1-xln(x))/(xe^x)

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

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

Solution steps

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

Expand

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

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

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

Simplify

= e^{- x} ln (x)

Add a constant to the solution

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

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