What is the integral of x^2e^{1/2 x} ?

The integral of x^2e^{1/2 x} is 2e^{x/2}x^2-16(1/2 e^{x/2}x-e^{x/2})+C

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integral of x^2e^{1/2 x}

\int x^{2} e^{\frac{1}{2} x} d x

2 e^{\frac{x}{2}} x^{2} - 16 (\frac{1}{2} e^{\frac{x}{2}} x - e^{\frac{x}{2}}) + C

Solution steps

\int x^{2} e^{\frac{1}{2} x} d x

Simplify

\frac{1}{2} x : \frac{x}{2}

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

Apply Integration By Parts

: 2 e^{\frac{x}{2}} x^{2} - \int 4 e^{\frac{x}{2}} x d x

= 2 e^{\frac{x}{2}} x^{2} - \int 4 e^{\frac{x}{2}} x d x

\int 4 e^{\frac{x}{2}} x d x = 16 (\frac{1}{2} e^{\frac{x}{2}} x - e^{\frac{x}{2}})

= 2 e^{\frac{x}{2}} x^{2} - 16 (\frac{1}{2} e^{\frac{x}{2}} x - e^{\frac{x}{2}})

Add a constant to the solution

= 2 e^{\frac{x}{2}} x^{2} - 16 (\frac{1}{2} e^{\frac{x}{2}} x - e^{\frac{x}{2}}) + C

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What is the integral of x^2e^{1/2 x} ?
The integral of x^2e^{1/2 x} is 2e^{x/2}x^2-16(1/2 e^{x/2}x-e^{x/2})+C