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

The integral of x^2e^{x/3} is 3e^{x/3}x^2-6(3e^{x/3}x-9e^{x/3})+C

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integral of x^2e^{x/3}

\int x^{2} e^{\frac{x}{3}} d x

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

Solution steps

\int x^{2} e^{\frac{x}{3}} d x

Apply Integration By Parts

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

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

Add a constant to the solution

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

What is the integral of x^2e^{x/3} ?
The integral of x^2e^{x/3} is 3e^{x/3}x^2-6(3e^{x/3}x-9e^{x/3})+C