What is the integral of 3x^2e^{x-1} ?

The integral of 3x^2e^{x-1} is 3(x^2e^{x-1}-2(e^{x-1}x-e^{x-1}))+C

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integral of 3x^2e^{x-1}

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

3 (x^{2} e^{x - 1} - 2 (e^{x - 1} x - e^{x - 1})) + C

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int x^{2} e^{x - 1} d x

Apply Integration By Parts

: x^{2} e^{x - 1} - \int 2 x e^{x - 1} d x

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

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

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

Add a constant to the solution

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

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