What is the integral of x^2e^xln(x) ?

The integral of x^2e^xln(x) is e^xx^2ln(x)-2e^xxln(x)+2e^xln(x)-e^xx+3e^x-2\Ei(x)+C

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integral of x^2e^xln(x)

\int x^{2} e^{x} ln (x) d x

e^{x} x^{2} ln (x) - 2 e^{x} x ln (x) + 2 e^{x} ln (x) - e^{x} x + 3 e^{x} - 2 Ei (x) + C

Solution steps

\int x^{2} e^{x} ln (x) d x

Apply Integration By Parts

: ln (x) (e^{x} x^{2} - 2 (e^{x} x - e^{x})) - \int \frac{e ^{x} x ^{2} - 2 ( e ^{x} x - e ^{x} )}{x} d x

= ln (x) (e^{x} x^{2} - 2 (e^{x} x - e^{x})) - \int \frac{e ^{x} x ^{2} - 2 ( e ^{x} x - e ^{x} )}{x} d x

\int \frac{e ^{x} x ^{2} - 2 ( e ^{x} x - e ^{x} )}{x} d x = e^{x} x - 3 e^{x} + 2 Ei (x)

= ln (x) (e^{x} x^{2} - 2 (e^{x} x - e^{x})) - (e^{x} x - 3 e^{x} + 2 Ei (x))

Simplify

ln (x) (e^{x} x^{2} - 2 (e^{x} x - e^{x})) - (e^{x} x - 3 e^{x} + 2 Ei (x)) : e^{x} x^{2} ln (x) - 2 e^{x} x ln (x) + 2 e^{x} ln (x) - e^{x} x + 3 e^{x} - 2 Ei (x)

= e^{x} x^{2} ln (x) - 2 e^{x} x ln (x) + 2 e^{x} ln (x) - e^{x} x + 3 e^{x} - 2 Ei (x)

Add a constant to the solution

= e^{x} x^{2} ln (x) - 2 e^{x} x ln (x) + 2 e^{x} ln (x) - e^{x} x + 3 e^{x} - 2 Ei (x) + C

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What is the integral of x^2e^xln(x) ?
The integral of x^2e^xln(x) is e^xx^2ln(x)-2e^xxln(x)+2e^xln(x)-e^xx+3e^x-2\Ei(x)+C