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

The integral of 2x^3e^{2x} is e^{2x}x^3-3/2 e^{2x}x^2+3/2 e^{2x}x-3/4 e^{2x}+C

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

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

e^{2 x} x^{3} - \frac{3}{2} e^{2 x} x^{2} + \frac{3}{2} e^{2 x} x - \frac{3}{4} e^{2 x} + C

Solution steps

\int 2 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

= 2 \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

= 2 (\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}))

= 2 (\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})))

Simplify

2 (\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}))) : e^{2 x} x^{3} - \frac{3}{2} e^{2 x} x^{2} + \frac{3}{2} e^{2 x} x - \frac{3}{4} e^{2 x}

= e^{2 x} x^{3} - \frac{3}{2} e^{2 x} x^{2} + \frac{3}{2} e^{2 x} x - \frac{3}{4} e^{2 x}

Add a constant to the solution

= e^{2 x} x^{3} - \frac{3}{2} e^{2 x} x^{2} + \frac{3}{2} e^{2 x} x - \frac{3}{4} e^{2 x} + C

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What is the integral of 2x^3e^{2x} ?
The integral of 2x^3e^{2x} is e^{2x}x^3-3/2 e^{2x}x^2+3/2 e^{2x}x-3/4 e^{2x}+C