What is the integral of 4x^3e^{2x} ?

The integral of 4x^3e^{2x} is 4(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 4x^3e^{2x}

\int 4 x^{3} e^{2 x} d x

4 (\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 4 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

= 4 \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

= 4 (\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}))

= 4 (\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

= 4 (\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

\int \frac{17.6 \cdot 15.24}{17.6 \cdot 0 ^{2} + 1}

x \to 0 lim (4 x^{2} + 3)

9 y^{^{''}} - 9 y^{^{'}} - 4 y = 0

\frac{d}{d x} (2 x^{4} - 6 x^{3} - 7 x^{2} + 9)

f (x) = 3 z (2 z^{2} - 7 z + 1)^{4}

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