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

The integral of 2e^{x/2}x^3 is 2(2e^{x/2}x^3-6(2e^{x/2}x^2-16(1/2 e^{x/2}x-e^{x/2})))+C

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

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

2 (2 e^{\frac{x}{2}} x^{3} - 6 (2 e^{\frac{x}{2}} x^{2} - 16 (\frac{1}{2} e^{\frac{x}{2}} x - e^{\frac{x}{2}}))) + C

Solution steps

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

\int 6 e^{\frac{x}{2}} x^{2} d x = 6 (2 e^{\frac{x}{2}} x^{2} - 16 (\frac{1}{2} e^{\frac{x}{2}} x - e^{\frac{x}{2}}))

= 2 (2 e^{\frac{x}{2}} x^{3} - 6 (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 (2 e^{\frac{x}{2}} x^{3} - 6 (2 e^{\frac{x}{2}} x^{2} - 16 (\frac{1}{2} e^{\frac{x}{2}} x - e^{\frac{x}{2}}))) + C

\frac{\partial}{\partial x} (e^{- r (x - c t)^{2}})

\frac{d}{d x} (\frac{3 - 5 x - x ^{2}}{x ^{2} - 6})

x \to 0 lim (\frac{x}{x ^{2} - 5 x})

x \to \infty + lim (x^{2})

y = x^{2} - 2 x - 4

What is the integral of 2e^{x/2}x^3 ?
The integral of 2e^{x/2}x^3 is 2(2e^{x/2}x^3-6(2e^{x/2}x^2-16(1/2 e^{x/2}x-e^{x/2})))+C