What is the integral of-x^5e^x ?

The integral of-x^5e^x is -e^xx^5+5e^xx^4-20e^xx^3+60e^xx^2-120e^xx+120e^x+C

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integral of-x^5e^x

\int - x^{5} e^{x} d x

- e^{x} x^{5} + 5 e^{x} x^{4} - 20 e^{x} x^{3} + 60 e^{x} x^{2} - 120 e^{x} x + 120 e^{x} + C

Solution steps

\int - x^{5} e^{x} d x

Take the constant out:

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

= - \int x^{5} e^{x} d x

Apply Integration By Parts

: x^{5} e^{x} - \int 5 x^{4} e^{x} d x

= - (x^{5} e^{x} - \int 5 x^{4} e^{x} d x)

\int 5 x^{4} e^{x} d x = 5 (x^{4} e^{x} - 4 (x^{3} e^{x} - 3 (x^{2} e^{x} - 2 (e^{x} x - e^{x}))))

= - (x^{5} e^{x} - 5 (x^{4} e^{x} - 4 (x^{3} e^{x} - 3 (x^{2} e^{x} - 2 (e^{x} x - e^{x})))))

Simplify

- (x^{5} e^{x} - 5 (x^{4} e^{x} - 4 (x^{3} e^{x} - 3 (x^{2} e^{x} - 2 (e^{x} x - e^{x}))))) : - e^{x} x^{5} + 5 e^{x} x^{4} - 20 e^{x} x^{3} + 60 e^{x} x^{2} - 120 e^{x} x + 120 e^{x}

= - e^{x} x^{5} + 5 e^{x} x^{4} - 20 e^{x} x^{3} + 60 e^{x} x^{2} - 120 e^{x} x + 120 e^{x}

Add a constant to the solution

= - e^{x} x^{5} + 5 e^{x} x^{4} - 20 e^{x} x^{3} + 60 e^{x} x^{2} - 120 e^{x} x + 120 e^{x} + C

\frac{\partial}{\partial x} (4 y x^{3} - 6 y^{2} x^{2})

f (x) = \frac{1}{1 + x ^{2}}, (1, \frac{1}{2})

\frac{d}{d y} (4)

\int 15 z^{3} e^{z} d z

e^{- 4 (t - 1)}

What is the integral of-x^5e^x ?
The integral of-x^5e^x is -e^xx^5+5e^xx^4-20e^xx^3+60e^xx^2-120e^xx+120e^x+C