What is the integral of (x^2+5x)e^{x/3} ?

The integral of (x^2+5x)e^{x/3} is 3e^{x/3}x^2-3e^{x/3}x+9e^{x/3}+C

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integral of (x^2+5x)e^{x/3}

\int (x^{2} + 5 x) e^{\frac{x}{3}} d x

3 e^{\frac{x}{3}} x^{2} - 3 e^{\frac{x}{3}} x + 9 e^{\frac{x}{3}} + C

Solution steps

\int (x^{2} + 5 x) e^{\frac{x}{3}} d x

Expand

(x^{2} + 5 x) e^{\frac{x}{3}} : e^{\frac{x}{3}} x^{2} + 5 e^{\frac{x}{3}} x

= \int e^{\frac{x}{3}} x^{2} + 5 e^{\frac{x}{3}} 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^{\frac{x}{3}} x^{2} d x + \int 5 e^{\frac{x}{3}} x d x

\int e^{\frac{x}{3}} x^{2} d x = 3 e^{\frac{x}{3}} x^{2} - 6 (3 e^{\frac{x}{3}} x - 9 e^{\frac{x}{3}})

\int 5 e^{\frac{x}{3}} x d x = 5 (3 e^{\frac{x}{3}} x - 9 e^{\frac{x}{3}})

= 3 e^{\frac{x}{3}} x^{2} - 6 (3 e^{\frac{x}{3}} x - 9 e^{\frac{x}{3}}) + 5 (3 e^{\frac{x}{3}} x - 9 e^{\frac{x}{3}})

Simplify

= 3 e^{\frac{x}{3}} x^{2} - 3 e^{\frac{x}{3}} x + 9 e^{\frac{x}{3}}

Add a constant to the solution

= 3 e^{\frac{x}{3}} x^{2} - 3 e^{\frac{x}{3}} x + 9 e^{\frac{x}{3}} + C

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What is the integral of (x^2+5x)e^{x/3} ?
The integral of (x^2+5x)e^{x/3} is 3e^{x/3}x^2-3e^{x/3}x+9e^{x/3}+C