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

The integral of (x+x^3)e^x is e^xx^3-3e^xx^2+7e^xx-7e^x+C

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integral of (x+x^3)e^x

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

e^{x} x^{3} - 3 e^{x} x^{2} + 7 e^{x} x - 7 e^{x} + C

Solution steps

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

Expand

(x + x^{3}) e^{x} : e^{x} x + e^{x} x^{3}

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

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

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

Simplify

e^{x} x - e^{x} + x^{3} e^{x} - 3 (x^{2} e^{x} - 2 (e^{x} x - e^{x})) : e^{x} x^{3} - 3 e^{x} x^{2} + 7 e^{x} x - 7 e^{x}

= e^{x} x^{3} - 3 e^{x} x^{2} + 7 e^{x} x - 7 e^{x}

Add a constant to the solution

= e^{x} x^{3} - 3 e^{x} x^{2} + 7 e^{x} x - 7 e^{x} + C

What is the integral of (x+x^3)e^x ?
The integral of (x+x^3)e^x is e^xx^3-3e^xx^2+7e^xx-7e^x+C