What is the integral of e^x(x^2-2x-1) ?

The integral of e^x(x^2-2x-1) is e^xx^2-4e^xx+3e^x+C

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integral of e^x(x^2-2x-1)

\int e^{x} (x^{2} - 2 x - 1) d x

e^{x} x^{2} - 4 e^{x} x + 3 e^{x} + C

Solution steps

\int e^{x} (x^{2} - 2 x - 1) d x

Apply Integration By Parts

= (x^{2} - 2 x - 1) e^{x} - \int (2 x - 2) e^{x} d x

\int (2 x - 2) e^{x} d x = 2 e^{x} x - 4 e^{x}

= (x^{2} - 2 x - 1) e^{x} - (2 e^{x} x - 4 e^{x})

Simplify

(x^{2} - 2 x - 1) e^{x} - (2 e^{x} x - 4 e^{x}) : e^{x} x^{2} - 4 e^{x} x + 3 e^{x}

= e^{x} x^{2} - 4 e^{x} x + 3 e^{x}

Add a constant to the solution

= e^{x} x^{2} - 4 e^{x} x + 3 e^{x} + C

What is the integral of e^x(x^2-2x-1) ?
The integral of e^x(x^2-2x-1) is e^xx^2-4e^xx+3e^x+C