What is the integral of (x^2+3)e^x ?

The integral of (x^2+3)e^x is e^xx^2-2e^xx+5e^x+C

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integral of (x^2+3)e^x

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

e^{x} x^{2} - 2 e^{x} x + 5 e^{x} + C

Solution steps

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

Apply Integration By Parts

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

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

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

Simplify

(x^{2} + 3) e^{x} - 2 (e^{x} x - e^{x}) : e^{x} x^{2} - 2 e^{x} x + 5 e^{x}

= e^{x} x^{2} - 2 e^{x} x + 5 e^{x}

Add a constant to the solution

= e^{x} x^{2} - 2 e^{x} x + 5 e^{x} + C

What is the integral of (x^2+3)e^x ?
The integral of (x^2+3)e^x is e^xx^2-2e^xx+5e^x+C