What is the integral of 2x^3e^x ?

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

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integral of 2x^3e^x

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

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

Solution steps

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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

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