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

The integral of x(3x^2-2)e^x is 3e^xx^3-9e^xx^2+16e^xx-16e^x+C

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

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

3 e^{x} x^{3} - 9 e^{x} x^{2} + 16 e^{x} x - 16 e^{x} + C

Solution steps

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

Expand

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

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

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

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

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

Simplify

3 (x^{3} e^{x} - 3 (x^{2} e^{x} - 2 (e^{x} x - e^{x}))) - 2 (e^{x} x - e^{x}) : 3 e^{x} x^{3} - 9 e^{x} x^{2} + 16 e^{x} x - 16 e^{x}

= 3 e^{x} x^{3} - 9 e^{x} x^{2} + 16 e^{x} x - 16 e^{x}

Add a constant to the solution

= 3 e^{x} x^{3} - 9 e^{x} x^{2} + 16 e^{x} x - 16 e^{x} + C

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What is the integral of x(3x^2-2)e^x ?
The integral of x(3x^2-2)e^x is 3e^xx^3-9e^xx^2+16e^xx-16e^x+C