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

The integral of e^{x/3}(x^2+2x) is 3e^{x/3}x^2-4(3e^{x/3}x-9e^{x/3})+C

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

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

3 e^{\frac{x}{3}} x^{2} - 4 (3 e^{\frac{x}{3}} x - 9 e^{\frac{x}{3}}) + C

Solution steps

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

Expand

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

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

Simplify

= 3 e^{\frac{x}{3}} x^{2} - 4 (3 e^{\frac{x}{3}} x - 9 e^{\frac{x}{3}})

Add a constant to the solution

= 3 e^{\frac{x}{3}} x^{2} - 4 (3 e^{\frac{x}{3}} x - 9 e^{\frac{x}{3}}) + C

What is the integral of e^{x/3}(x^2+2x) ?
The integral of e^{x/3}(x^2+2x) is 3e^{x/3}x^2-4(3e^{x/3}x-9e^{x/3})+C