What is the integral of-x^4e^x ?

The integral of-x^4e^x is -e^xx^4+4e^xx^3-12e^xx^2+24e^xx-24e^x+C

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integral of-x^4e^x

\int - x^{4} e^{x} d x

- e^{x} x^{4} + 4 e^{x} x^{3} - 12 e^{x} x^{2} + 24 e^{x} x - 24 e^{x} + C

Solution steps

\int - x^{4} e^{x} d x

Take the constant out:

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

= - \int x^{4} e^{x} d x

Apply Integration By Parts

= - (x^{4} e^{x} - \int 4 x^{3} e^{x} d x)

\int 4 x^{3} e^{x} d x = 4 (x^{3} e^{x} - 3 (x^{2} e^{x} - 2 (e^{x} x - e^{x})))

= - (x^{4} e^{x} - 4 (x^{3} e^{x} - 3 (x^{2} e^{x} - 2 (e^{x} x - e^{x}))))

Simplify

- (x^{4} e^{x} - 4 (x^{3} e^{x} - 3 (x^{2} e^{x} - 2 (e^{x} x - e^{x})))) : - e^{x} x^{4} + 4 e^{x} x^{3} - 12 e^{x} x^{2} + 24 e^{x} x - 24 e^{x}

= - e^{x} x^{4} + 4 e^{x} x^{3} - 12 e^{x} x^{2} + 24 e^{x} x - 24 e^{x}

Add a constant to the solution

= - e^{x} x^{4} + 4 e^{x} x^{3} - 12 e^{x} x^{2} + 24 e^{x} x - 24 e^{x} + C

What is the integral of-x^4e^x ?
The integral of-x^4e^x is -e^xx^4+4e^xx^3-12e^xx^2+24e^xx-24e^x+C