What is the integral of (e^x)/x (1+xln(x)) ?

The integral of (e^x)/x (1+xln(x)) is e^xln(x)+C

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integral of (e^x)/x (1+xln(x))

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

e^{x} ln (x) + C

Solution steps

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

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

Expand

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

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

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

\int e^{x} ln (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 (e^x)/x (1+xln(x)) ?
The integral of (e^x)/x (1+xln(x)) is e^xln(x)+C