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

The integral of ln(x)e^x is e^xln(x)-\Ei(x)+C

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integral of ln(x)e^x

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

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

Solution steps

\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)

Add a constant to the solution

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

\int \frac{3 - x}{x ^{2} + 1} d x

\frac{\partial}{\partial x} (\frac{5 x y ^{2}}{x ^{2} + y ^{2}})

\frac{d}{d x} (x^{2} 3^{x})

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

y = f (x) = cos (3 x^{2})

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