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

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

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integral of e^x(x^2+2)

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

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