What is the integral of (x^2-x+5)e^x ?

The integral of (x^2-x+5)e^x is e^xx^2-3e^xx+8e^x+C

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integral of (x^2-x+5)e^x

\int (x^{2} - x + 5) e^{x} d x

e^{x} x^{2} - 3 e^{x} x + 8 e^{x} + C

Solution steps

\int (x^{2} - x + 5) e^{x} d x

Expand

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

= \int e^{x} x^{2} - e^{x} x + 5 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^{2} d x - \int e^{x} x d x + \int 5 e^{x} d x

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

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

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

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

Simplify

x^{2} e^{x} - 2 (e^{x} x - e^{x}) - (e^{x} x - e^{x}) + 5 e^{x} : e^{x} x^{2} - 3 e^{x} x + 8 e^{x}

= e^{x} x^{2} - 3 e^{x} x + 8 e^{x}

Add a constant to the solution

= e^{x} x^{2} - 3 e^{x} x + 8 e^{x} + C

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\int \frac{x y}{2} d x

\int \frac{1}{x - 2 x ^{2}} d x

\int 2 x ln^{2} (x) d x

What is the integral of (x^2-x+5)e^x ?
The integral of (x^2-x+5)e^x is e^xx^2-3e^xx+8e^x+C