What is the integral of e^x(5-x)^3 ?

The integral of e^x(5-x)^3 is -e^xx^3+18e^xx^2-111e^xx+236e^x+C

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integral of e^x(5-x)^3

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

- e^{x} x^{3} + 18 e^{x} x^{2} - 111 e^{x} x + 236 e^{x} + C

Solution steps

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

Expand

e^{x} (5 - x)^{3} : 125 e^{x} - 75 e^{x} x + 15 e^{x} x^{2} - e^{x} x^{3}

= \int 125 e^{x} - 75 e^{x} x + 15 e^{x} x^{2} - e^{x} x^{3} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 125 e^{x} d x - \int 75 e^{x} x d x + \int 15 e^{x} x^{2} d x - \int e^{x} x^{3} d x

\int 125 e^{x} d x = 125 e^{x}

\int 75 e^{x} x d x = 75 (e^{x} x - e^{x})

\int 15 e^{x} x^{2} d x = 15 (x^{2} e^{x} - 2 (e^{x} x - e^{x}))

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

= 125 e^{x} - 75 (e^{x} x - e^{x}) + 15 (x^{2} e^{x} - 2 (e^{x} x - e^{x})) - (x^{3} e^{x} - 3 (x^{2} e^{x} - 2 (e^{x} x - e^{x})))

Simplify

125 e^{x} - 75 (e^{x} x - e^{x}) + 15 (x^{2} e^{x} - 2 (e^{x} x - e^{x})) - (x^{3} e^{x} - 3 (x^{2} e^{x} - 2 (e^{x} x - e^{x}))) : - e^{x} x^{3} + 18 e^{x} x^{2} - 111 e^{x} x + 236 e^{x}

= - e^{x} x^{3} + 18 e^{x} x^{2} - 111 e^{x} x + 236 e^{x}

Add a constant to the solution

= - e^{x} x^{3} + 18 e^{x} x^{2} - 111 e^{x} x + 236 e^{x} + C

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What is the integral of e^x(5-x)^3 ?
The integral of e^x(5-x)^3 is -e^xx^3+18e^xx^2-111e^xx+236e^x+C