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

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

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

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

Apply Integration By Parts

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

= 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 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 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{d}{d x} ((3 x^{2} + 7)^{3})

x \to + 2 - lim (x (x - 1))

\int \frac{2 x ^{3}}{( x ^{2} + 1 ) ^{2}} d x

\frac{d}{d x} (\frac{( x + 1 )}{( x ^{3} + x - 2 )})

\frac{2 x}{x ^{2} + 1}

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