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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

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

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

Simplify

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

Add a constant to the solution

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

x \to 3 lim (4 x^{2} - \frac{1}{x})

\int \frac{2 x ^{3} + 7 x ^{2} + 2 x + 9}{2 x + 3} d x

x e^{- x^{10}}

y^{^{'}} - 2 y = - 4 y^{3}

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

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