What is the integral of 6x^3e^{2x} ?

The integral of 6x^3e^{2x} is 6(1/2 e^{2x}x^3-3/2 (1/2 e^{2x}x^2-1/4 (e^{2x}*2x-e^{2x})))+C

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integral of 6x^3e^{2x}

\int 6 x^{3} e^{2 x} d x

6 (\frac{1}{2} e^{2 x} x^{3} - \frac{3}{2} (\frac{1}{2} e^{2 x} x^{2} - \frac{1}{4} (e^{2 x} \cdot 2 x - e^{2 x}))) + C

Solution steps

\int 6 x^{3} e^{2 x} d x

Take the constant out:

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

= 6 \cdot \int x^{3} e^{2 x} d x

Apply Integration By Parts

: \frac{1}{2} e^{2 x} x^{3} - \int \frac{3}{2} e^{2 x} x^{2} d x

= 6 (\frac{1}{2} e^{2 x} x^{3} - \int \frac{3}{2} e^{2 x} x^{2} d x)

\int \frac{3}{2} e^{2 x} x^{2} d x = \frac{3}{2} (\frac{1}{2} e^{2 x} x^{2} - \frac{1}{4} (e^{2 x} \cdot 2 x - e^{2 x}))

= 6 (\frac{1}{2} e^{2 x} x^{3} - \frac{3}{2} (\frac{1}{2} e^{2 x} x^{2} - \frac{1}{4} (e^{2 x} \cdot 2 x - e^{2 x})))

Add a constant to the solution

= 6 (\frac{1}{2} e^{2 x} x^{3} - \frac{3}{2} (\frac{1}{2} e^{2 x} x^{2} - \frac{1}{4} (e^{2 x} \cdot 2 x - e^{2 x}))) + C

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What is the integral of 6x^3e^{2x} ?
The integral of 6x^3e^{2x} is 6(1/2 e^{2x}x^3-3/2 (1/2 e^{2x}x^2-1/4 (e^{2x}*2x-e^{2x})))+C