What is the integral of (1+3x+x^2)e^x ?

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

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integral of (1+3x+x^2)e^x

\int (1 + 3 x + x^{2}) e^{x} d x

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

Solution steps

\int (1 + 3 x + x^{2}) e^{x} d x

Expand

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

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

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

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

Simplify

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

Add a constant to the solution

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

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