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

The integral of 5x^2e^{x/3} is 5(3e^{x/3}x^2-6(3e^{x/3}x-9e^{x/3}))+C

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integral of 5x^2e^{x/3}

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

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

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 5 \cdot \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

= 5 (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}})

= 5 (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

= 5 (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 5x^2e^{x/3} ?
The integral of 5x^2e^{x/3} is 5(3e^{x/3}x^2-6(3e^{x/3}x-9e^{x/3}))+C