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

The integral of 1/2 x^2e^{-1/2 x} is -e^{-x/2}(x^2+4(x+2))+C

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

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

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

Solution steps

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

Take the constant out:

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

= \frac{1}{2} \cdot \int x^{2} e^{- \frac{1}{2} x} d x

Simplify

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

= \frac{1}{2} \cdot \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

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

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

Simplify

\frac{1}{2} (- 2 e^{- \frac{x}{2}} x^{2} - (- 4 (- 2 e^{- \frac{x}{2}} x - 4 e^{- \frac{x}{2}}))) : - e^{- \frac{x}{2}} (x^{2} + 4 (x + 2))

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

Add a constant to the solution

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

x^{2} e^{- x} + (ln (x))^{2}

\int 3 s h^{3} x d x

\frac{70}{t}

y^{^{'}} = 5 x

\int (\frac{3}{x ^{4}} + 2 - \frac{3}{x ^{2}}) d x

What is the integral of 1/2 x^2e^{-1/2 x} ?
The integral of 1/2 x^2e^{-1/2 x} is -e^{-x/2}(x^2+4(x+2))+C