What is the integral of (x^3+5x-1)e^x ?

The integral of (x^3+5x-1)e^x is e^xx^3-3e^xx^2+11e^xx-12e^x+C

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integral of (x^3+5x-1)e^x

\int (x^{3} + 5 x - 1) e^{x} d x

e^{x} x^{3} - 3 e^{x} x^{2} + 11 e^{x} x - 12 e^{x} + C

Solution steps

\int (x^{3} + 5 x - 1) e^{x} d x

Expand

(x^{3} + 5 x - 1) e^{x} : e^{x} x^{3} + 5 e^{x} x - e^{x}

= \int e^{x} x^{3} + 5 e^{x} x - e^{x} 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 e^{x} x^{3} d x + \int 5 e^{x} x d x - \int e^{x} d x

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

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

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

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

Simplify

x^{3} e^{x} - 3 (x^{2} e^{x} - 2 (e^{x} x - e^{x})) + 5 (e^{x} x - e^{x}) - e^{x} : e^{x} x^{3} - 3 e^{x} x^{2} + 11 e^{x} x - 12 e^{x}

= e^{x} x^{3} - 3 e^{x} x^{2} + 11 e^{x} x - 12 e^{x}

Add a constant to the solution

= e^{x} x^{3} - 3 e^{x} x^{2} + 11 e^{x} x - 12 e^{x} + C

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What is the integral of (x^3+5x-1)e^x ?
The integral of (x^3+5x-1)e^x is e^xx^3-3e^xx^2+11e^xx-12e^x+C