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

The integral of (3x^2-1)e^x is 3e^xx^2-6e^xx+5e^x+C

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

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

3 e^{x} x^{2} - 6 e^{x} x + 5 e^{x} + C

Solution steps

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

Apply Integration By Parts

= (3 x^{2} - 1) e^{x} - \int 6 x e^{x} d x

\int 6 x e^{x} d x = 6 (e^{x} x - e^{x})

= (3 x^{2} - 1) e^{x} - 6 (e^{x} x - e^{x})

Simplify

(3 x^{2} - 1) e^{x} - 6 (e^{x} x - e^{x}) : 3 e^{x} x^{2} - 6 e^{x} x + 5 e^{x}

= 3 e^{x} x^{2} - 6 e^{x} x + 5 e^{x}

Add a constant to the solution

= 3 e^{x} x^{2} - 6 e^{x} x + 5 e^{x} + C

What is the integral of (3x^2-1)e^x ?
The integral of (3x^2-1)e^x is 3e^xx^2-6e^xx+5e^x+C