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

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

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

\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}}) + C

Solution steps

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

Apply Integration By Parts

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

= 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

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