Question

In Exercises \(21-24,\) use tabular integration to find the antiderivative. $$\int x^{3} e^{-2 x} d x$$

Short answer

The antiderivative of \(x^3 e^{-2x}\) with respect to \(x\) is \(-\frac{1}{2}x^{3}e^{-2x} - \frac{3}{4}x^{2}e^{-2x} - \frac{3}{4}x e^{-2x} - \frac{3}{8} e^{-2x} + C\).

Step-by-step solution

Choose the function to differentiate and the function to integrate

To start, choose the function to be differentiated and the one to be integrated in the tabular integration. In this case, \(x^3\) is selected to differentiate because its derivative will eventually become zero after several differentiations. On the other hand, \(e^{-2x}\) is chosen to integrate because it does not change much when integrated, it's easy to integrate indefinitely.

Create the tabular integration table

Next, create a table for the tabular integration. In the left column, start with the function to differentiate at the top and compute its successive derivatives until zero is reached. On the right column, start with the function to integrate and compute its successive integrations. Alternate plus/minus signs beginning with a plus at the top. The table becomes:\n \[ \begin{{array}}{{c c|c}} + & x^3 & e^{-2x} \ - & 3x^2 & -\frac{1}{2}e^{-2x} \ + & 6x & \frac{1}{4}e^{-2x} \ - & 6 & -\frac{1}{8}e^{-2x} \ + & 0 & \frac{1}{16}e^{-2x} \end{{array}} \]

Multiply diagonally and sum

For the final step, each term in the left column of the table is multiplied by the term diagonally below it or to its right in the right column. Add or subtract these products according to the signs in the left column. The result is the antiderivative (up to a constant) of the original integrand:\n \[ \int x^3 e^{-2x} dx = x^3 \cdot (-\frac{1}{2}e^{-2x}) - 3x^2 \cdot (\frac{1}{4}e^{-2x}) + 6x \cdot (\frac{-1}{8}e^{-2x}) - 6 \cdot (\frac{1}{16}e^{-2x}) \]

Simplify the result

Finally, you simplify the result to give: \[ \int x^3 e^{-2x} dx = -\frac{1}{2}x^{3}e^{-2x} - \frac{3}{4}x^{2}e^{-2x} - \frac{3}{4}x e^{-2x} - \frac{3}{8} e^{-2x} + C \] where \(C\) is the constant of integration.