What is the integral of x^2e^{ln(x)+x} ?

The integral of x^2e^{ln(x)+x} is x^3e^x-3(x^2e^x-2(e^xx-e^x))+C

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integral of x^2e^{ln(x)+x}

integral

x^{2} e^{l n (x) + x}

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

Solution steps

\int x^{2} e^{l n (x) + x} d x

Simplify

x^{2} e^{l n (x) + x} : e^{x} x^{3}

= \int e^{x} x^{3} d x

Apply Integration By Parts

= x^{3} e^{x} - \int 3 x^{2} e^{x} d x

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

= x^{3} e^{x} - 3 (x^{2} e^{x} - 2 (e^{x} x - e^{x}))

Add a constant to the solution

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

What is the integral of x^2e^{ln(x)+x} ?
The integral of x^2e^{ln(x)+x} is x^3e^x-3(x^2e^x-2(e^xx-e^x))+C