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

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

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

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

e^{x} ln (x) + C

Solution steps

\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

\frac{\partial}{\partial y} (7 e^{x} sin (y))

f (x) = - \frac{5}{x ^{3}}

x \to - 3 lim (\frac{3 + x}{x})

\int \frac{1}{2 y} d y

e^{2 x}

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